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RANDOM MATRIX ENSEMBLES WITH SOFTCONFINEMENT POTENTIAL By JINMYUNG CHOI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2010 2010 Jinmyung Choi To my family ACKNOWLEDGMENTS First of all, I would like to thank my advisor, Khandker Muttalib for all the help. I would also like to thank him for all the valuable discussions, which led me to understand various aspects of random matrix theory and to develop my interests around the subject. Second, I would like to thank my family and friends for their constant love and encouragement. And lastly, I would like to thank Department of Physics at University of Florida for offering me the valuable teaching experiences. TABLE OF CONTENTS ACKNOWLEDGMENTS .......................... LIST O F FIG URES .. .. .. .. .. .. .. ABSTRACT ................... ......... CHAPTER 1 M OTIVATIO N . . 1.1 Random Matrix Theory and Complex Systems ..... 1.2 Complex Systems with Powerlaw Distributions ..... 1.3 Fattail RM Models ..................... 1.3.1 Levy Matrices .................. 1.3.2 Free Levy Matrices ................. 1.3.3 Nonextensive q Ensembles .. .......... 1.3.4 Ensembles Based on Superpositions of Classical Ensem bles . . 2 RANDOM MATRIX THEORY .. ................ 2.1 Random Matrix Ensembles .. .............. 2.2 Orthogonal Polynomial Method .......... .... 2.2.1 Joint Probability Distribution Function (JPDF) .. 2.2.2 Determinant Form of JPDF and Kernel ...... 2.2.3 Correlation Functions and Cluster Functions ... 2.2.4 Unfolding . . 2.3 Gaussian Ensembles .. ................ 2.3.1 Sem iCircle law .................. 2.3.2 Sine Kernel . . 2.3.3 Gap Probability and Spacing Distribution ..... 2.3.4 Number Variance .................. 2.4 Critical Ensem bles .. .................. 2.4.1 Anderson Transition and Critical Ensembles . 2.4.2 Critical Statistics . . 2.5 Coulomb Gas Analogy ................... 3 AENSEMBLES ............... ............ 3.1 M ethod . . 3.2 R results . . 3.2.1 Ageneralization of qpolynomials ......... 3.2.2 Eigenvalue Density ................. 3.2.3 Twolevel Correlation Function ............ page . 4 or Gaussian 3.2.3.1 Observation of Normal/Anomalous Structure of Twolevel Kernel . .. .. 50 3.2.3.2 Twolevel Kernel of Aensembles .... 52 3.2.3.3 Universality of Aensembles .... 57 3.2.4 Number Variance ...... ..... ........ ....... .. 60 3.3 D discussion . .. 60 4 CO NC LUSIO N . . 63 APPENDIX A CENTRAL LIMITTHEOREM .......... ...... ........... .. 65 B NDEPENDENCE PROBLEM IN SPREAD FUNCTION APPROACH 68 R EFER EN C ES . . 71 BIOGRAPHICAL SKETCH ................... ............. 75 LIST OF FIGURES Figure 31 Log Rn as a function of n for different values of A. Solid line corresponds to the critical ensemble A = 1 ........................ 32 The exponent a(A) as a function of A for two different values of 7 In(1/q). A = 1 corresponds to the qpolynomials describing the critical ensembles. . 33 Density of eigenvalues for different values of A. .............. 34 Eigenvalue density (x = 0.50), 0.5 (7 = 0.25) 35 Eigenvalue density (x = 0.50), 0.5 (7 = 0.25) 36 Eigenvalue density (x = 2.00), 1.5 (7 = 4.00) 37 Eigenvalue density (x = 2.00), 1.5 (7 = 4.00) 38 Eigenvalue density for (7 = 0.25) as well as 1 39 Eigenvalue density for = 0 104) for Aensembles for A as well as 1 (7 = 0.75). ..... = 0 40) for Aensembles for A as well as 1 (7 = 0.75) ..... = 0 104) for Aensembles for A as well as 1 (7 = 0.75). ..... = 0 40) for Aensembles for A as well as 1 (7 = 0.75). Aensembles for A = 0 (7 = 0.75). ........ Aensembles for A = 1 (7 = 4.00) as well as 1 (7 = 0.75) .. ..... 310 Normal component of the cluster function for 0.5 as well as 1.0 (7 = 0.50) ....... 311 Normal component of the cluster function for 1.3 as well as 1.0 (7 = 1.50) ....... 312 Anomalous component of the cluster function 0.8, 07 as well as 1.0 (7 = 0.50) ..... 0.9 (7 = 0.75), 0.7 (7 0.9 (7 = 0.75), 0.7 (7 1.1 (7 = 0.75), 1.3 (7 1.1 (7 = 0.75), 1.3 (7 .9 (7 = 0.75), 0.7 ( = .1 (7 = 0.75), 1.3 ( = Aensembles for A = C Aensembles for A = 1 for Aensembles for A 313 Anomalous component of the cluster function for Aensembles for A 0.50), 0.5 2.00), 1.5 ).9, 0.7, .1, 1.2, =0.9, =1.1, 1.2, 1.3 as well as 1.0 (7 = 1.50). ........................... 314 Fitting results for normal component of the cluster function for Aensembles for A = 0.9, 0.7, 0.5 as well as 1.0 (7 = 0.50) ................ 315 Fitting results for normal component of the cluster function for Aensembles for A = 1.1, 1.2, 1.3 as well as 1.0 (7 = 1.50) ................ 316 Fitting results for anomalous component of the cluster function for Aensembles for A =0.9, 0.8, 07 as well as 1.0 (7 = 0.50). ................... . = ( : t 317 Fitting results for anomalous component of the cluster function for Aensembles for A =1.1, 1.2, 1.3 as well as 1.0 (7 = 1.50). .... 59 318 Number variance for Aensembles for A =0.5, 0.8, 1,0, 1.5, 2.0 (7 = 0.5). 61 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy RANDOM MATRIX ENSEMBLES WITH SOFTCONFINEMENT POTENTIAL By Jinmyung Choi August 2010 Chair: Khandker Muttalib Major: Physics In this work, we study invariantclass of random matrix ensembles characterized by the asymptotic logarithmic softconfinement potential V(H) ~ [In H](1+') (A > 0), named "Aensembles". The suggestion is inspired by the existing random matrix models such as the critical ensembles (A=1), the free Levy matrices (A 0 limit) and the Gaussian ensembles (A oo limit) in an effort to investigate the novel universality associated with the fattail random matrix ensembles as well as the logarithmic softconfinement potential within the framework of rotationally invariant random matrix theory. First of all, we show that the orthogonal polynomials with respect to the weight function exp[(In x)1+'] belong to a novel orthogonal polynomial system, named "Ageneralization of qpolynomials". Second, we show that based on numerical construction of the "Ageneralization of qpolynomials", we can study the onelevel and the twolevel correlation functions as well as the level statistics of the Aensembles. Third, we show that the onelevel correlation (eigenvalue density) has a powerlaw form p(x) oc [In x]I1/x and the unfolded twolevel correlation function possesses the normal/anomalous structure, characteristic of the critical ensembles. We further show that the anomalous part, socalled "ghostcorrelation peak" is controlled by the parameter A; decreasing A increases the anomaly. Third, we also identify the twolevel kernel of the Aensembles in the semiclassical regime, which can be written in a sinhkernel form with more general argument that reduces to that of the critical ensembles for A = 1. Forth, we show that the number variance is linear in L for all A and the slope (the level compressibility) is increasing as A decreases, which is consistent with the Adependence of sum rule violation 0 < X(A) < 1. Finally, we will discuss the novel universality of the Aensembles, which interpolates the Gaussian ensembles (A oo limit), the critical ensembles (A = 1), the free Levy matrices (A + 0 limit). CHAPTER 1 MOTIVATION 1.1 Random Matrix Theory and Complex Systems Random matrix theory (RMT) deals with eigenvalue and eigenvector correlations of random matrix ensembles drawn in a stochastic manner. The earlier application of RMT [1, 2] in physics dates back to the 1950s when Wigner introduced RMT to explain the fluctuation properties of heavy nuclei energy spectrum. In a slow nuclear reaction, incident nucleon interacts with the constituents of target nucleus in a complicated manner such that extra energy carried in by the an incident nucleon is shared nontrivially with the nucleons of the target nucleus while forming the excited states. However, the fluctuation properties of the excitation energy levels of the compound nucleus are not very well understood [3]. To explain the statistical nature of the energy spectrum of the compound nucleus, Wigner suggested to consider ensembles of Hamiltonians whose entries are randomly drawn from the Gaussian distribution. It turns out that the statistical properties of the eigenvalue correlations of such random Hamiltonian ensembles, e.g. the Gaussian ensembles, showed a good agreement with the statistical behaviors of the eigenvalue spectrum obtained from various heavy nuclei reaction experiments [3, 4]. The underlying idea of RMT is well explained by Dyson [5]: What is required is a new kind of statistical mechanics, in which we renounce exact knowledge not of the state of the system but of the system itself. We picture a complex nucleus as a black box in which a large number of particles are interacting according to unknown laws. As in the orthodox statistical mechanics we shall consider an ensemble of Hamiltonians, each of which could describe a different nucleus. There is a strong logical expectation, though no rigorous mathematical proof, that an ensemble average will correctly describe the behavior of one particular system which is under observation. The expectation is strong, because the system might be one of a huge variety of systems, and a very few of them will deviate much from a properly chosen ensemble average. One of the surprising aspects of this "new kind of statistical mechanics" is that its utility is far reaching beyond its origination. Since its introduction to the manybody complex nuclei systems, RMT has been applied to a wide variety of systems in diverse areas, including manybody atoms and molecules, quantum chaos, mesoscopic disordered conductor, 2D quantum gravity, conformal field theory, chiral phase transitions as well as zeros of Riemann zeta function, scalefree networks, biological networks, communication systems and financial markets [618]. This broad range of applicability of RMT in seemingly unrelated areas highlights the universal features of the correlations of the eigenvalues in RMT. Within the classical or Gaussian model pioneered by Wigner, these correlations are known as the WignerDyson (WD) statistics of the Gaussian ensembles, which are qualitatively different from the statistical features of completely uncorrelated eigenvalues given by the Poisson statistics. 1.2 Complex Systems with Powerlaw Distributions In the past decades, there have been growing interest of RMT applications to generic complex systems that appear in the study of natural, social, economical and biological systems. In these systems, the microscopic interactions among the constituents are so complicated that the statistical hypothesis made in RMT seems relevant. Namely, we are forced to "renounce exact knowledge not of the state of the system but of the system itself." Thus, it appears to be appropriate to apply the idea of RMT to understand certain statistical features of these systems. One such application of RMT is found in the analysis of the empirical covariance matrices in a multivariate setting such as financial assets and climate systems [11, 12, 19] based on classical RM models. In these works, the application aims at decomposing genuine correlations, present in the eigenvalues and eigenvectors of the empirical covariance matrices from pure noise components, the RMT predictions. For example, Refs. [11, 12] study the empirical crosscorrelation matrices of the stock price returns of S&P 500 and the largest 1000 US companies for certain period time such that the elements of such matrices represent the correlations among the companies in a given period, e.g., S (r(t)r(t)). (11) Here < ... > denotes time average over the period studied and r,(t) represents a normalized time series of stock price return of a company i defined as ( R(t) (R,(t)) (12) r,(t) (12) where the volatility (R,(t)2) (Ri(t))(R(t)) and R,(t) represents the logarithm of the price change, S,(t A t) S,(t) Ri(t) Int A) t + t) InS(t) (13) si(t) Here, Si(t) is the time series of the stock price. Thus, the covariance matrix C, is by construction a real symmetric matrix. The statistical properties of the covariance matrix C. can be compared to the well known results of appropriate classical RM models, e.g. Gaussian orthogonal ensembles (GOE) or Wishart matrices (Laguerre ensembles). The study shows that most of the eigenvalues except some outliers are well within the RMT expectations in terms of the eigenvalue density, the nearest spacing distribution, the number variance and the inverse participation ratio, implying that the empirical covariance matrices are dominated by random Gaussian noise for the most part. The small fraction of the outliers carries some meaningful informations useful for the risk management. This explanation, however, is not fully satisfactory because the a priori assumption that crosscorrelations present in the stock price returns are dominated by the Gaussian noise is not in accordance with the observation that the distribution of stock price returns follows an asymptotic fattail distribution, the so called inversecubic law [20]. To be precise, it is shown in Ref. [20] that the cumulative distribution of the stock price returns has an asymptotic fattail distribution, e.g. P(ri(t) > r) ~ ra with a 3. Thus, the application of the classical Gaussian RM models requires reconsideration in this example and more generally, in the complex systems where the fluctuation is better characterized by fattail noises. This situation motivates search for suitable RM models that can incorporate the fattail noises. In this context, many attempts have been made to construct generalized random matrix ensembles that incorporate powerlaw or fattail distributions [2128]. The significance of such generalization beyond the Gaussian ensembles is mainly two fold. First, there are numerous complex systems that exhibit fattail noises, notably, financial markets, earthquakes, scalefree networks etc. [2932]. As pointed out earlier, the relevance of the classical RM models in these systems seems questionable. It is because the classical or Gaussian RM models are based on the assumption that the systems are characterized by the Gaussian noise, which is not appropriate for systems with fattail noise where the occurrence of extreme events are not as rare as expected from normal distributions. Second, it is conceived that the universality of classical or Gaussian RM models is closely linked to the prevalence of normal or Gaussian distributions in nature, a consequence of the central limit theorem [21]. It has, however, not been fully investigated if there is a counterpart of the Gaussian ensemble as implied by the structure of the generalized central limit theorem [21], comparing e.g, the Gaussian and the Levy basins (refer to APPENDIX A for more details). So far, fattail distributions in random matrix ensembles have been carefully incorporated in some limited cases and the calculation of the correlation functions of the eigenvalues have been carried out for certain special cases [2128]. However, the question regarding the universality of the correlations of the eigenvalues remains unresolved. For Gaussian ensembles, it is the well known twolevel sine kernel that establishes the universality of the correlations in the properly scaled large N (matrix size) limit but it is not clear if there exists a similar universal twolevel kernel for the powerlaw or the fattail ensembles as well. 1.3 Fattail RM Models 1.3.1 Levy Matrices The Levy matrices [21] are introduced by P. Cizeau and J.P. Bouchaud in consideration of constructing a RM model based on the Levy probability distribution. In particular, they considered a Nx N real symmetric matrix ensemble of independent indenticallydistributed (i.i.d.) random numbers of asymptotic Levy distributions given by the following: P(H) H" O <<2 (14) IHU.I1+p , where Hy denotes the matrix elements. By construction, H, = Hji and Ho is set to be N11/ in order to ensure that the typical largest element of a row is 0(1). Since the matrix elements are drawn from the distributions with divergent moments, there is no direct way to construct an analytical model within the standard RMT techniques. Thus, they developed a novel analytical technique based on the cavity method [21] to evaluate the eigenvalue density and the inverse participation ratio in parallel with numerical simulations. The main results of their work are summarized in the following: 1. The eigenvalue density converges to a limiting form of distribution in the large N limit that has asymptotic powerlaw distribution. e.g. p(x oo) ~ which has the same scaling power as that of the matrix elements. 2. From careful examinations of the inverse participation ratio, it is shown that the eigenstates undergo a nontrivial localizationdelocalization transition within the eigenvalue spectrum at a certain critical value of x = x depending on the parameter p: for p > 1, there exists a critical value x, that differentiate the extended states (x < x) from the algebraically localized states (x > xc), which is an unusual type of localization since this type of localization allows nonzero conductivity. For p < 1, all the states are localized (finite participation ratio) except ground states. The states below x, are localized in a usual sense. However, the finite fraction of the states above x, are still extended over O(N) site. In the limit p  1 or 2, the x, diverges. 3. The numericallyobtained nearest neighbor spacing distribution is found to be nonuniversal 1, which depends on the location of the eigenvalue spectrum considered. 1.3.2 Free Levy Matrices The free Levy matrices, introduced by Z. Burda et al. [22], are constructed based on the theory of free random variable (FRV) that offers the probability theory of noncommuting variables as a generalization of the classical probability theory. The correspondence between the classical probability theory and the FRV theory is made in Ref. [33]: 1. Probability distribution P(x) : Spectral density p(x) = ImG(x + iO)/7 2. Characteristic function P(k) : Green function G(z) < Tr(z ') > 3. Logarithm of characteristic function In P(k) : Rtransform R(G(z)) z  In the classical probability theory, the logarithm of characteristic function is additive (the additivity of the cumulants) under convolution operation of two random variables. Analogously, in the FRV theory, the Rtransform of two independent random matrixvalued numbers, M, and M2, is additive, i.e., RMI+M2(Z) = RMI(z) + RM2(Z) [22, 33]. The free Levy random matrix approach takes advantage of the fact that one can reconstruct the Green function, and the spectral density, and finally identify a particular probability measure that characterizes a random matrix ensemble from the known Rtransforms that correspond to all free stable probability distributions under FRV calculus: e.g., R(z) = a + bza where 0 < a < 2 (ca 4 1)2 which determines the asymptotic powerlaw behavior of the stable distributions. The parameters a and b are 1 It should be emphasized that the numerical unfolding procedure is nontrivial in this model so that it requires further investigation regarding the universality of the spacing distribution. In fact, our numerical investigation shows possible universality of the spacing distribution. 2 For a = 1, R(z) = a i'(1 +) 2 In yz. real valued and are associated with the shift, the slope (a) and the skewness (3) and the range (7) of the stable distributions. Note that R(z) is given without any particular realization of matrix ensembles. In order to find a particular realization of a random matrix ensemble characterized by the probability measure of the form eNTr V(M) dM (15) where Tr is the trace and V(M) is specific to the realization of the ensembles and is related to the confining potential in the Coulomb gas analogy of the classical RM models developed by Dyson [34], Z. Burda et al. [22] first calculated the spectral densities 3 from the stable Rtransforms for some exactly solvable cases of parameters. They showed that the asymptotic form of spectral densities displays powerlaw distributions with the exponents within the Levy stability regime. They then showed that in all such cases, the asymptotic form of the confining potential V(x) is given as V(x) = In x2 + O(1/x) by using the relation between the potential and the spectral density arising in the Coulomb gas analogy. After the identification of the confining potential and thus the matrix ensemble (free Levy matrices), they further studied the eigenvalue correlations based on a standard RMT technique, namely the orthogonal polynomial method. This model successfully incorporates the asymptotic Levy distributions and also allows the calculations of eigenvalue correlations within the standard RMT techniques due to the rotational invariance of the probability measure of the ensemble. It, however, suggests that the 3 To give a quick illustration of the method, consider the trivial choice of R(z) = a, then Green function G(z) = a and thus the spectral density p(x) = 6(x a). Refer to Ref [22] for more details. twolevel correlations of the ensembles have a nontrivial Ndependence that cannot be simply scaled out. 1.3.3 Nonextensive q Ensembles One way of constructing the classical RM models is to maximize the Shannon entropy S S = dHP(H)In P(H), (16) subject to the condition that the probability is normalized to 1, I dHP(H) = 1. (17) Here dH is the measure associated with the matrix elements of H. In particular, requiring that the variance of the matrix elements be finite, (equivalently assuming that extremely large matrix elements are improbable) J dHTr(H2)P(H) < o, (18) it can be shown by using the Lagrange multiplier (A) method that the P(H) that maximizes the entropy S subject to the the above conditions is given by P(H)dH eTrH2 dH. (19) Thus, the maximization of the entropy with the finitevariance constraint on P(H) leads to the Gaussian ensembles. In a similar manner, the nonextensive q ensembles [23] can be constructed from maximizing the nonextensive entropy Sq S[P(H)] f dH[P(H) (110) q1 with the usual normalization condition, f dH P(H) = 1 and q is a parameter. Similar to the constraint that requires the finite variance of P(H) leading to the Gaussian ensembles, the nonextensive q ensembles require the following condition f dH trH2 [P(H)] 2 (111) f dH [P(H)] with a constant a. Using the Lagrange multiplier method, it can be shown that the P(H) that maximizes the nonextensive entropy Sq subject to the above constraints is given by P(H) ~ expq(AtrH2) (112) where the qexponential function exp,(x) is defined as exp,(x) {[1 (1 q)x]+}, (113) with [...]+ = max(..., 0). (114) Note that for q=1, the exp,(x) reduces to the usual exponential function thus leading to the usual Gaussian ensembles. The study shows that the nonextensive q ensembles exhibit characteristically different behavior depending on the parameter q; for q > 1, the distributions of eigenvalue density show true long tails and for q < 1, the distributions have compact support. However, the variability of the parameter q in this case depends on the dimensionality N of the ensemble such that in the large N limit where universal behavior is expected, the maximum q allowed for the nonextensive ensembles approaches unity. 1.3.4 Ensembles Based on Superpositions of Classical or Gaussian Ensembles There are several RM models categorized under this class of ensembles [24 28]. The main idea is to construct generalized ensembles with the superposition (or deformation) of the Gaussian or Wishart ensembles. To illustrate the details of the idea, the approach in Ref. [24] will be discussed in the following. The underlying idea of the approach in the Ref. [24] is to work with an associated characteristic function, rather than a probability distribution, to discover a novel random matrix ensemble. It is shown in this framework that the general npoint correlation function can be written down as dbf(b) Rn(x , ,xn) = (4b) n/2det[K (;xi, )],i 1,2,...n (115) Jo (4b) /2N' where f(b) is called the spread function, defined as any nonnegative normalizable function, f f(b)db = 1 and ;x = xi/2v b and KN refers to the well known sine kernel of the Gaussian ensembles. In particular, the spectral density (1point correlation function) can be given by UN(X) = j dbf(b) KG(x,x) (116) 27 Jo b/2 and two point cluster function defined as T2(x1, x2) R2(x, x2) + R1(x1)R1(x2) has the form T2 = T20 T2, where f dbf(b)[KN( ) 2 2 T(xix2) = [ ,) (117) and f" dbf(b) 4b TN(x ) Nb(x) ON(XN(x2) (118) Thus, a variety of possible generalized ensembles can be obtained from a proper choice of superposition (a choice of the spread function f(b)) of the kernel of Gaussian ensemble. At the same time, it is possible to study the correlation functions for the choice of f(b) immediately. For example, the Gaussian ensembles can be thought of as a trivial example corresponding to the choice of f(b) = 6(b bo). For a nontrivial choice of f(b) = b(C+ 1e b, it is shown that the spectral density exhibit asymptotic powerlaw distributions. Although this framework allows a successful realization of RM model that can incorporate the desired fattail spectral density, it is not yet clear if the unfolded twolevel cluster function can have Nindependent asymptotic limit 4 4 It turns out that the twolevel correlation function carries nontrivial N dependence. The details will be provided in APPENDIX B. CHAPTER 2 RANDOM MATRIX THEORY 2.1 Random Matrix Ensembles The first step to construct a RME is to consider a set of N x N Hermitian matrices H with the following probability measure PN(H)dH oc eTrV(H)dH, (21) where V(H) is a suitably increasing function such that the probability measure is normalizable. Tr is the matrix trace and dH the invariant measure. Especially, the TrV(H) ensures the rotational invariance under orthogonal, unitary and symplectic transformations. More explicitly, it means that under a transformation R, H  M = RtHR, RtR = 1 (22) the probability measure remains the same PN (H)dH = PN(M)dM. (23) Thus, by construction, all orientations of the eigenbasis are equally likely or in other words, there is no preferential basis. Each of the symmetries determines the structure of the hamiltonian systems, e.g., * Orthogonal symmetry (timereversal invariant systems with rotational symmetry) Real symmetric matrices Unitary symmetry (systems in which timereversal symmetry is broken) Complex Hermitian matrices Symplectic symmetry (timereversal invariant systems with halfinteger spin and broken rotational symmetry) Selfdual quaternion matrices. In order to find a specific realization of a RME, one needs to specify the probability measure V(H) of a RME. One way to find V(H) comes from the maximization of the entropy or the minimization of information content of the RME. Suppose that nothing is known about the detailed dynamics of a system other than the fundamental symmetries, which gives no knowledge about the matrix elements of the hamiltonian systems other than the global symmetries. The information (I) contained in the N x N Hamiltonian matrix H can be defined as I dHP(H)InP(H), (24) with the condition that the probability is normalized to 1, dHP(H) = 1. (25) requiring the variance of the matrix elements be finite I dHTr(H)P(H) < o. (26) The information in Eq. 24 can be minimized subject to the constraints in Eq. 25 and Eq. 26 by using Lagrange multiplier method. For an arbitrary variation 6P(H) of P(H), 61 = dHP(H) { In P(H) ATr(H2)P(H)} = 0 (27) Thus, P(H)dH oc eT'(H2)dH. (28) This is the probability measure of the wellknown Gaussian ensembles. The Gaussian ensembles can also be obtained based on the two assumptions; invariance under transformation (Orthogonal, Unitary, Symplectic) and statistical independence of matrix elements [4]. The three different symmetries define three different classes of Gaussian ensembles: Gaussian Orthogonal Ensemble (GOE), Gaussian Unitary Ensemble (GUE), and Gaussian Symplectic Ensemble (GSE). If we only require the rotational invariance of P(H)dH, dropping the assumption of the statistical independence of the matrix elements, then P(H) may be any function of the traces of powers of H. Thus, in general the probability measure can be written down as P(H)dH oc eTrV(H)dH. (29) It is also possible to consider RMEs with noninavarant form of probability measure under transformation, which is the socalled "Wigner class" ensembles [35]. In this class, each matrix element of the ensemble of N x N matrix are randomly taken from i.i.d. probability distributions so that the generic form of the probability measure of Wigner ensembles can be given by P(M)dM oc f(Mi) H dMi (210) ij ij where f(x) is the probability distribution of each individual matrix element and the product n[i is performed for all the degrees of the freedom of the ensemble under consideration. Some of the examples are: * Wigner Guassian ensembles : Hermitian matrices whose entries are given by i.i.d. Gaussian distributions. Levy matrices (Wigner Levy matrices) : Real symmetric matrices whose entries are given by i.i.d. Levy distributions. Banded matrices : Symmetric matrices whose entries m, are nonzero for ijl < r and zero otherwise where r determines band width. Adjacency or Laplacian matrices of random graphs or scalefree networks. In the following, we will mainly focus on the study of the invariant class ensembles and its universality. 2.2 Orthogonal Polynomial Method The orthogonal polynomial method has been a fruitful tool in the development of RMT. The merit of the orthogonal polynomial method is that it allows to write down all the correlation functions of the eigenvalues in terms of orthogonal polynomials corresponding to the probability measure characterizing a RME. In particular, the twolevel correlation function written in a properly scaled variable in the asymptotic large N limit is central to establishing the universality of RMEs. Thus, in the following, we will review the orthogonal polynomial method and the results obtained for classical and Gaussian RMEs and the critical ensembles. 2.2.1 Joint Probability Distribution Function (JPDF) In order to study the eigenvalue correlations, the probability measure P(H)dH needs to be transformed from the matrix element basis to the eigenvalue/eigenvector basis, H RtXR with RtR = 1. (211) Here, the matrix X is the diagonal matrix containing eigenvalues ({x,}, i = 1 ..., N) and the matrix R is the rotational transformation containing eigenvectors. After the transformation is considered, the invariant measure dH in the matrix element basis can be rewritten in the eigenvalue/eigenvector basis, e.g., dH = J(X, R)dXdR; i = 1... N (212) where J(X, R) is the Jacobian of the transform. The dX and dR refer to the measure associated with the eigenvalues and parameters of the rotational transformations respectively. Here, the Jacobian factor can be further calculated for a given symmetry class and it can be shown that [4] J(X,R) = ]x xj, f(R) (213) i where the 3 is the symmetry parameter of the rotational transformations such as Orthogonal (3 = 1), Unitary (3 = 2) and Symplectic (3 = 4) symmetries. The f(R) is the function of the parameters associated with the rotational transformations. Once the parameters associated with transformation is integrated out, we obtain the JPDF of N eigenvalues in the following form. PN({x,})dX oc I Ix, X w({x,})dx ... dxN (214) i where n, function. For Gaussian ensembles, w(x) = exp( x2). For more general potential V(x), w(x) = exp( V(x)). 2.2.2 Determinant Form of JPDF and Kernel The form of JPDF obtained in Eq. 214 can be greatly simplified by using matrix determinant. In particular, the unitary class (3 = 2) is the simplest and thus, in the following we will focus on the unitary case of JPDF. To this end, we first observe that the Vandermonde determinant n,< Ix j = det[x'l]. More explicitly, 1 1 ... 1 X1N1 X2 ... XN1 i After absorbing the weight function into the determinant above, JPDF can be written down as PN({x,}) = det[QTQ] (216) ZN where ZN is a normalization constant and the matrix Q is defined as 1 1w(xw(2) ... W) 1W(X) W(X1) X2 (2 ... XNW(X) (217) xN1 /wMw( x'\ N1 w( N1 w/w N2 ViWX2) ... XN / W(XN) Since the determinant does not change by the adding to any one row or column to multiples of the other rows or columns, the matrix Q can be rewritten as po(x) V (x P1(xi) w(x1) po(x2) wX2) p1(x2) w(X2) (... N)p (X) .. N(XN) WXN) PN 1(X1)wI V X1) PN1(X2)V ) ...w PN1(XN)VW(XN) where pi(x) denotes an arbitrary polynomial of degree i. And matrix multiplication QTQ, we can obtain 1) = PN({xi}) = Idet N! KN(xl, x1) KN (2, x 1) KN (xN, x1) KN(X1, x2) KN (X2, x2) KN (XN, X2) after considering the ... KN(x, xN) ... KN(x2, xN) ... KN(xN, xN) where the twolevel kernel KN(x, y) is defined as N1 KN(X, y) ,i(x)i(y), i=o where the "wave functions" /bi(x) pi(x) (x = pi(x)ev(x)2 Here the choice pi(x) is arbitrary but if pi(x) is chosen a system of the polynomials that are orthogonal with respect to the weight function w(x) = ev(x), i.e., Pn(x)pm(X)W(x)dx = 6mn. The kernel KN(x, y) satisfies following important properties. SKN(x, x)dx = N and KN(x, z) = K(X, y)KN(y, z)dy. (218) (219) (220) (221) (222) (223) These imply that / det[KN(xi, y)] ij... mdxm = (N m+ 1)det[KN(xi, ,x)],J i... m (224) 2.2.3 Correlation Functions and Cluster Functions The probability of finding any n eigenvalues out of the available N eigenvalues in the intervals, {x, + x, + dx,; i= 1... n} is given by nlevel correlation function Rx,...x) = N I dxn ,... dxN P(x (X, .. ). (225) (N I n)!J J Using the determinant form of JPDF and the properties of the kernel, the nlevel correlation function can be written in a compact form given by R,(x,, x, ...., x,) = det[KN(xi, x)]{,(i= ...,n}. (226) Note that the diagonal terms of the determinant are given by the onelevel correlation function Ri(x) (or the eigenvalue density oaN(x)) N1 R (x) = KN(X, x) >= i(x) i(y). (227) i= and the offdiagonal terms of the determinant are given by twolevel kernel in terms of which the twolevel correlation function can be written down as R2(x, y) = KN(x,x)KN(y, y) KN(x, y)KN(y,x). (228) For practical purpose, it is useful to introduce the nlevel cluster functions defined by T, = (1)n '( 1)! n RG (x, with in Gk) (229) G i=1 Here, G stands for any division of the indices (1, 2, ....n) into m subgroups (Gi, G2,..., Gm). For example, the one level cluster function T7 (x) = R1 (x) (230) and twolevel cluster function T2(x, y) = R2(x, y) + R(x, x) R(y, y) (231) = KN(X, y)2 (232) N 1 x ( ) (2 3 3 ) 2.2.4 Unfolding Within the invariant class ensembles, the onelevel correlation function (eigenvalue density) aN(x) = KN(x, x) is not universal and it is dependent on the specific form of the probability measure dP(H). So in the limit of large matrix size, e.g., N oo, we need to introduce a proper scaling variable to study the universality of correlation function such that in the scaled variable the mean spacing of eigenvalues become unity. It is achieved by defining the unfolding variable du lim KNv(x,x)dx. (234) N>oo The unfolded twolevel kernel can be written in terms of the unfolding variables ((u, v) = lim KN() (235) N VKAN(X,x)KN(y, y) where u and v are the unfolding variables. Now all the correlation functions and the cluster functions can be written in terms of this unfolded kernel. For example, unfolded onelevel cluster functions Y(u) = 1 and the two level cluster function Y2(u, v) = K(u, v)2 (236) Thus, the study of correlation functions requires understanding of two level kernel in the properly scaled variable which requires the knowledge of asymptotic behavior of the orthogonal polynomials. Note that if the kernel satisfy Eq. 223 or in order words, the kernel is reproducing, the unfolded twolevel kernel satisfies the normalization of sum rule. I dvY(u, v) K K(x, y)K lim j KN(y, y) K(x,y)2 dy Nrn KN(x, x)KN(y, y) lim Kx,y)Ky,x)dyy Noo KN (x, x) n KN (x, x) limN =K 1. Noo KN(x, x) 2.3 Gaussian Ensembles 2.3.1 SemiCircle law For the Gaussian ensembles V(x) wellknown semicircle law. o(x) = KN(x, x)= x2, the eigenvalue density is given by the i(x) (x) M /2N x2 (240) where the wave function ,',,(x) = hn(x)ex2/2 satisfies 0o ,',(x), (y)e 2dx = vF22nn!nm (241) where hn(x) is the Hermite polynomial of degree n. The derivation of semicircle law can be given in the following way. First of all, we recognize that Eq. 219, the determinant form of the JPDF, can be considered as the probability density of the ground state of the manybody wave function of noninteracting fermions, e.g., P(xi, ..., XN) ox I (x1, ..., xN) 2 where bo (X2) b l(Xl) (242) Y" (X1, ..., X ) (237) (238) (239) bo (xi) bi(Xi) ... ^o(XV) ... bl (Xl) (243) Since the wave function in the determinant is the solution of onedimensional harmonic oscillator, it satisfies the Schrodinger equation Thus, ( d2 1 h dX2 + '.,(x) = (n + 1/2)h,.,(x). (244) In the large n = N 1 limit, the Fermi momentum can be read off locally at x PF = /2N 1 x2. (245) The density of one dimensional fermions is related to the Fermi momentum PF by 1 Pp PF S(x) ldp= h=l. (246) 27hx) J p2 7 Combining Eq. 245 and Eq. 246, we get o(x) v 2NV x2. (247) 7 2.3.2 Sine Kernel In order to calculate the unfolded two level kernel, we consider the spectrum around the origin in the large N limit, socalled double scaling limit (x 0 and N oo limit) such that xv/N is finite. The proper unfolded variable in this limit du = KN(x, x)dx = d2Ndx. Using the definition of the kernel, N1 KN(u, v)= lim ( u)i( ). (248) Recalling the ChristoffelDarboux formula, i(x) (y) = (249) 2 x y = 0 (u, ) = l im (250) N ooV2 u V Using asymptotic solution for harmonic oscillator wave function, lim (1)"m 1/4 2m( ) 1/2 COs(u) (251) mmoo 2Ni lim (1)m 1/4 2m+l ) = 1/2 sin(7u) (252) moo 2/V where N = 2m. We get the unfolded kernel G(u sin(w(u v)) (253) KG (u, V)= (253) v (u v) which is the wellknown sine kernel. 2.3.3 Gap Probability and Spacing Distribution The probability that there is no eigenvalue in the interval of s is called gap probability defined as a Fredholm determinant E(s) = det[1 G] (254) where the kG refers to the sine kernel over the interval of s/2 and s/2. The spacing distribution is related to the gap probability by d2 E(s) p(s) =ds (255) which is commonly used to study short range fluctuations in the eigenvalue spectrum. This function measures the probability of finding two neighboring eigenvalues in the interval of s in the unfolded scale. The analytical calculation is highly nontrivial but the good approximation of the spacing distribution can be obtained from considering 2 x 2 matrix model suggested by Wigner. The general expression for the spacing distribution, Wigner's surmise or WignerDyson statistics, p(s) = cs ed2 (256) where cp and d3 are constants. The 3 refers to the symmetry parameter of the Gaussian ensembles. Note that in the small s limit, the spacing distribution is determined by s1 that indicates that the two neighboring levels repel. The level repulsion is characteristic of the Gaussian ensembles. This is in contrast to the uncorrelated or Possion case, p(s) = es, in which the level repulsion is absent and therefore the levels can be bunched up or separated far apart. 2.3.4 Number Variance The number variance E(s) = (n2) (n)2 provides the measure of the longrange eigenvalue fluctuation. It is given by j s/2 s/2 Z(s) = [R(u, v)]dudv (257) Js/2 Js/2 /s2 s/2 = s2[(u v) Y(u,v)]dudv (258) Js/2 Js/2 where R(u, v) is twolevel correlation function defined by R(u, v) (p(u)p(v)) 1. Using the cluster function of GUE, Y(u, v) = YG( v) = [sn(v)) the further calculation gives ZG(s) = s2 (s r)Y(r)dr (259) = (In(27s)+ 7 + 1) O(s1) (260) where 7 is Euler's constant, 7 = 0.5772.... Note that for large s, EG(s) o In s, which displays another characteristic of the Gaussian Ensembles, the level rigidity. In contrast, for the Poisson or uncorrelated case, RP(u, v) = 6(u v) and thus EP(s) = s. 2.4 Critical Ensembles 2.4.1 Anderson Transition and Critical Ensembles The suggestion of a novel universality beyond the Gaussian ensembles comes from the study of the Anderson transition in the disordered electronic systems[8, 3642]. In these systems, the Gaussian ensemble is only relevant in the metallic regime where all the eigenstates are extended across the entire system and correlations of the corresponding eigenvalues are well described by the WD statistics. As the disorder is made strong enough, the eigenstates become localized and thus the eigenvalues become uncorrelated. Especially at the delocalizationlocalization transition, it has been established that the correlations of the eigenvectors exhibit novel features [8, 3642] such as multifractality and the correlations of the eigenvalues lead to a level compressibility that is intermediate between WD and Poisson statistics. Similarly in the studies of quantum chaos, energy level statistics of systems that are intermediate between chaotic and regular states also require generalization beyond WD and Poisson statistics [3, 4347]. In these contexts, extensive studies have been carried out to construct a parametric generalization of RM models that cross over from WD to Poisson [3, 8, 3647] as a function of the parameter. Some of these generalizations indeed capture the essential features of the critical statistics, among which the family of qRMEs [4850] provides a particularly valuable insight. Within the common framework of rotationally invariant RM models [4] the qRMEs show how the universality of the Gaussian ensemble characterized by the zero parameter twolevel sine kernel breaks down and eventually gives rise to a different kind of universality for the critical ensembles, characterized by a oneparameter twolevel sinh kernel. In particular, the rotationally invariant RM models are characterized by a "confining potential" which defines the weight function of a set of orthogonal polynomials; the key difference between the Gaussian and the critical ensembles comes from the fact that the corresponding orthogonal polynomials, namely classical vs. qorthogonal polynomials, respectively, possess qualitatively different asymptotic properties [51, 52]. 2.4.2 Critical Statistics The conjecture of the novel universality at the Anderson transition is based on the studies of the spacing distribution at the critical point of the tight binding hamiltonian on a cubic lattice (L x L x L) given by H= Z ,c c, c c, (261) where c] and c, are the creation and annihilation operators at site i, andj denotes the neighbor sites of i thusj = 1,..., 6 on the cubic lattice; ci is the random energy of the site / in unit of overlap energy of neighboring sites that is uniformly distributed in the range [ W/2, W/2]. In particular, Ref. [36] showed that the spacing distribution Pc(s) is scaleinvariant (does not depend on the system size L) at the transition point W = We by investigating the quantity 7(W, L) 1; the sclaeinvariant Pc(s) is characterized by a linear slope in s for s < 1 and an exponential decay for s > 1. Thus the novel form of P(s) is hybrid of the WD statistics and the Poission statistics. Another characteristic of the critical statistics is the finite "level compressibility" or subPoissinan number variance, namely ZC(s) = Xs with 0 < X < 1, which is intermediate between the WD statistics EG(s) oc Ins and Poisson one EP(s) oc s. The origin of this behavior is known to be the sum rule violation. e.g., d(E(s)) L/2 00 X = d(s lim R(s)ds =1 Y(s) 0. (262) dS Loo jL12 O which is also related to the multifractal nature of the wave function correlation at the critical point [8, 3642]. 2.5 Coulomb Gas Analogy Dyson suggested that the JPDF of the invariant ensembles can be written as a form of Gibbs distribution for a classical onedimensional system of N particles described by 1 where A f2 P(s)ds and lower limit of the integral 26 refers to the crossing point of P,(s) and Pw(s) occurring at 2 6 2.002 the Hamiltonian H({xi}) [53] in the following form. 1 P({x,}) = Z exp[PH({x,})], (263) where H({x}) = In Ix, x + V(xi). (264) In this view point, the eigenvalues x, can be considered fictitious particles interacting with each other through logarithmic repulsion at temperature 1/3 while they are confined by the external potential V(x). In particular, one can assume that the particle density p(x) = "N 6(x x) reaches a continuum in the large asymptotic system size N oo limit. In this limit, the Hamiltonian H({xi}) can be read as an functional of the density. H[p] = I j dxdyp(x)p(y) In x y + dxp(x) V(x) (265) where J denotes the support of the density. Using the saddle point approximation, we can obtain an integral equation for the average particle (meanfield) density of (p(x)) 2 I dx(p(x)) In y x = V(y) + const. (267) where the additional constant term can be determined by the normalization condition of the density fp(x)dx = N. The two level correlation function can be written in terms of the functional derivative of (p(x)) with respect to V(x), 2 (p(x)e V(y)Dp (p(x)) = fx)e Dp (266) fJ e1H[p] Dp Thus, by taking functional derivative 6/6V(y) in Eq. 267, we get Ijdx(p(x))(p(y))R(x, z) In y z = 6(x y). (269) In the large N limit, around the region under consideration, if (p(x)) scales as N and becomes a constant, we can introduce a new variable u = x/A scaled by the mean level spacing A = p1 (i.e., for the Gaussian ensembles, p(x) oc 2N in the x 0 limit) and rewrite Eq. 269, which reads / dwR(u w) In v w = 0 6(u v). (270) This implies that the twolevel correlation function does not depend on the specific form of V(x), which carries microscopic information of the system. Therefore, the twolevel kernel is universal. Within this framework, it is understood that for the soft confinement potential, the average density is not constant in the N  oc limit so that it does not simply scale out. Therefore, the twolevel correlation function is expected to be different from the WD universality [54]. CHAPTER 3 AENSEMBLES In the motivation to investigate the universality associated with fattail or powerlaw RMEs, we introduced a family of U(N) invariant random matrix ensembles characterized by an asymptotic logarithmic potential V(H) = A[In H]1+X with A > 0 [55, 56], named "Aensembles" 1 The reason for such suggestion is based on the following few observations. First, it is known that for V(H) oc [In H]2 corresponding to A = 1 limit (the critical ensemble), the eigenvalue spectrum is given by inverse powerlaw distribution, which is known by the meanfield theoretic approach [57]. Second, for V(H) N In H corresponding to the A + 0 limit with the constant A being order of N (free Levy matrices), the spectral density is given by the fattail distributions. Third, in the limit A > 1, it is expected that the confinement potential may grow sufficiently strong, thereby approaching the Gaussian limit. Therefore, we can speculate that the parameter A is a controlling parameter of the powerlaw behavior. The fact that such parametric generalization (generalizing the power of the logarithm to arbitrary real value larger than 1) connects the existing RM models equipped with rotational invariance is interesting since the model allows us to explore any possible novel universality associated with fattail RMEs as well as the logarithmic softconfinement potential within the framework of the rotationally invariant RMT. The generic choice of the confining potential V(x) that gives asymptotic logarithmic behavior is V(x) = A[Inx]1+'. However, it has a unphysical singularity at the origin so that we need to regularize it in certain way. One possible way to do it is choosing, i.e., V(x) = A[In(l + x)]'+' but there are a variety of other forms that differ by the regularization behavior in the vicinity of origin, which will not change the characteristics 1 The earlier name 'Levy like ensembles' in Ref. [55] reflects the motivation of the study. of the Aensembles. For our study, we particularly choose the following form of the potential, 1 V(x) = [sinh x] +; A >0; q < 1. (31) In(1/q) For simplicity, we will introduce 7 In(1/q) so that the model has two parameter A and 7. The merit of choosing the Eq. 31 is that in the limit A = 1, it coincides with the one possible form of the weight function of the qRMEs so that we can compare our results with those of qRMEs. For the qRMEs, the mathematical properties of the corresponding orthogonal polynomials, known as "the IsmailMasson qpolynomials" [52], are well established and accordingly, the twolevel kernel (sinh kernel) and the all the spectral properties (the critical statistics) are very well known [48, 49]. In the following sections, we will show that for A 4 1, we can construct the corresponding orthogonal polynomials, "Ageneralization of qpolynomials" in a rigorous numerical method and thus we can study the onelevel (spectral density) and the twolevel kernel that is central to the test of the universality. We will also show that the spectral density and the twolevel correlation function of the A ensembles exhibit novel feature; the spectral density is given by a powerlaw form and twolevel kernel possess the anomalous component which is considered one of the characteristics of the critical ensembles. In addition, we will suggest a novel form of twolevel kernel of the Aensembles based on the unfolding analysis and discuss the details of its behavior as well as the properties of the level statistics that can be deduced from the kernel. Finally, we will discuss the implications and applications of the our results. 3.1 Method The main difficulty of studying Aensembles is that the orthogonal polynomials corresponding to the weight function w(x) = ev(x) for the arbitrary A values (expect A = 1) are not known. Thus, the first task is to obtain the orthogonal polynomials corresponding to the weight function of the Aensembles. In the following, we will review the procedure to construct orthogonal polynomials for nontrivial arbitrary weight function. To this end, we define the orthogonal polynomials O(x) for arbitrary weight function w(x), given by dxw(x)(Ox)n(xm) = 6n,mhn (32) where hn is the normalization constant. It is well known that every orthogonal system of real valued polynomials satisfy a three term recurrence relation [51] 2 X n(X) = 0n +(x) + SnOn(X) + n Rl_(x), (33) where S, and Rn are the real coefficients of the recurrence relation. In particular, Rn is related to the normalization constant hn by hn+ = Rn+ hn For example, the Hermite polynomials are determined by the recurrence relation of ,n+1(x) = xn(X) nn1(x). (34) The recurrence relation of the qpolynomials is given by 1 0n+ 1(xq) = X n(Xlq) q (1 qn)~nl(xlq), 0 < q < 1 (35) where q = e7, 7 > 0. We point out that for these cases, the weight function is an even function, namely V(x) = V(x) and thus, all Sn = 0. Therefore, the Rn determines the properties of the orthogonal polynomials. In particular, by comparing the Hermite polynomials and qpolynomials, we observe that Rn oc n (Hermite polynomials) (36) oc e"n (qpolynomials). (37) 2 Here, we consider that the On(x) is a monic polynomial. The significance of Rn is that it determines the upper bound of spectral density DN oc n/R and thus scaling behavior of the bulk of the spectrum in the large N limit. For example, for the Gaussian ensembles characterized by V(x) = x2 (Hermite polynomials), the upper bound of the spectral density is /N in the large N limit. Thus, the normalization condition of the spectral density requires the bulk of the spectrum to grow at an order of /N as in the semicircle law. While for the logarithmic softconfinement potential, the spectral edge grows at an exponential rate e"nl' (which will be shown in the next section), the bulk of the spectrum does not scale as N. i.e., for V(x) oc [In x]2, the spectral edge grows at ve, the bulk of the spectrum is given by p(x) oc 1/x which do not depend on N. One way to determine the orthogonal polynomials for an arbitrary weight function is to use the GramSchmidt determinant formula, e.g., a0 ,1 ... an a,1 2 ... an+1 1 n (X) (38) Gn1 an1 an ... a2n1 1 X ... Xn where the Gn stands for the GramSchmidt determinant. a0 a1 ... an a1 a2 ... ,:n Gn = a a2 an+ (39) an an ... a2n and a, are the moments given by a, = xw(x)dx (310) J/ Then, the Rn can be determined in terms of Gn by [58] GnGon2 Gn Gn, 2(311) 'n1 An alternative view of the above procedure is adopted in Ref. [59] to obtain the polynomials recursively. Following the Ref. [59], we define Qn,m xmbn(x)w(x)dx (312) 00 Using the fact that x" = Qn(x) + j aj@(x), we can find Qn, n= hn, (313) n1 Qn,n+l = hn S,, (314) J=0 and Qn,m = Qnl,m+l SnlQnl,m RnlQn2,m. (315) In this case, the determination of Rn and Sn in order to calculate the polynomials of degree n < N 1 requires only the knowledge of the 2N 1+ integrals of Qo,m (or the moments) Qo,m = xmw(x)dx (316) 00 for m = 0,.... 2N. 3.2 Results 3.2.1 Ageneralization of qpolynomials For the study of the Aensembles, we adopted the latter approach to construct the orthogonal polynomials. So first, we calculated Qo,m with Mathmatica Qom xmev(x)dx, (317) 00 with the V(x) shown in the Eq. 31. Since we choose V(x) to be symmetric around the origin, e.g., V(x) = V(x), the Sn = 0 for all n. Rn alone defines the corresponding 1e+25 I I I I k=0.6  / k=0.8  le+20 =1.0 = 1 .2 ................ k=1.8  / le+15 c) le+10 100000 1 e 0 5 '*' le05 5 10 15 20 25 n Figure 31. Log Rn as a function of n for different values of A. Solid line corresponds to the critical ensemble A = 1. polynomials. As described earlier, we can determine Rn with 2N + 1 integrals of Qo,m recursively. It turns out that the Rn obtained in this way shows an intriguing behavior that is depicted in Fig 31. After careful examination, we found that the behavior of Rn for large n should be of the form Rn oc e (318) where a(A) (319) The fitting result for a(A) is provided in Fig. 32. As A = 1, the a(A) is equal to 1 and thus In Rn grows linear in n, which coincides with the wellknown Rn behavior for the qpolynomials (see Eq. 36) and for A # 1, In Rn grows in n1'. We recognized that this is a novel behavior thus named the orthogonal polynomials of the Aensembles as Ageneralizaton of qpolynomials, which is one of the central results of our work. Note that this is dramatically different from that for all Freudlike y=0.5 A y=1.0 o 1.5  0.5 0 1] 0.5 1 I I I I 0 0.5 1 1.5 2 Figure 32. The exponent a(A) as a function of A for two different values of 7 In(l/q). A = 1 corresponds to the qpolynomials describing the critical ensembles. classical orthogonal polynomials whose weight function is given by w(x) = ex' with m > 0 where Rn oc n. 3.2.2 Eigenvalue Density The density of the eigenvalues p(x) = KN(x, x) can now be obtained for different values of A from Eq. 220 by summing the products numerically. The results are shown in Figure 33. Earlier it was understood in Ref. [55] that the density of the Aensembles is given by a pure powerlaw, e.g., p(x) = i For A = 1, 0 = 0 and for A > 1 and A < 1, 0 > 0 and 0 < 0. However, careful investigation shows that the earlier interpretation is only approximate and more accurate form of the eigenvalue density is given by [In x]^1 p(x) oc I for x > A (320) X where the lower cutoff A is dependent on the regularization of the confining potential V(x). i.e., for the choice of V(x) [In x]2, the singularity, albeit unphysical, can be extended in the vicinity of x 0 limit [60]. For our choice Eq. 31, such singularity 10 =0.6  k=1.0 S=, X=1.8  0.1 0.01 " .o 0.001  0.0001 le05 1e06 1 10 100 1000 10000 100000 le+06 log x Figure 33. Density of eigenvalues for different values of A. doesn't exist. In general, it is expected that the regularization behavior (the cutoff A) is different for the choice of V(x) sharing the same asymptotic logarithmic behavior. The following facts convince us that the spectral density should be given by Eq 320. First of all, it can be shown [61] that for A > 1, the density should be of the form 320 based on the meanfield approach, which reproduces the exact inverse powerlaw density as well as the sinh kernel of the critical ensembles (A = 1.) Second, the validity of this form for all A > 0 can be checked by considering the normalization condition of the spectral density 2 p(x)dx= N. (321) Here the factor 2 comes from the fact that p(x) is symmetric around origin. The upper bound DN is given by the largest zero of the orthogonal polynomials of order N, namely DN oc vRN. As pointed out, R oc exp[nl^]. We notice that Ref. [62] studied the largest zeros of the orthogonal polynomials to the weight function of exp[c(Inx)"] for c > 0 and m a positive even integer and reported that it is of order exp(nm1), which is the exact same behavior as the coefficient Rn of recurrence relation of the generalized qpolyonomials in Eq. 318. Thus, our results seem to imply that the results of Ref. [62] can be extended to an arbitrary real A > 0. Eq. 320 can be verified in the numerical calculation as well. However, since our V(x) is regularized near the origin unlike the choice V(x) = [In x]1+' used in the meanfield approach [60] and in Ref. [62], we expect that the exact form of p(x) will show agreement only for large x limit. To investigate this, we consider that the density is given by p(x) = (322) x+A where f(x; A) is a logarithmically slowly varying function and A is a constant arising due to regularization of the density at the origin. Thus, we can expect xp(x) to behave in the following way. X xp(x) = f(x; A), (323) x+A X f(x; A) x>A, (324) A Sf(x; A) x Fig. 34 and Fig. 35 show this behavior for A < 1 (A = 0.5, 0.7, 0.9, and 1) and the Fig. 36 and Fig. 37 for A > 1 (A = 1.1, 1.3, 1.5, andl) respectively. For all the cases, we chose y = 0(1), which ensures the cutoff A = 0(1). To further investigate if f(x; A) oc [In x]A1 for large x (x > 1), we plotted In[xp(x)] vs. In In x and fitted it in the range of 10 < x < 104. Fig 38 and Fig. 39 show the expected linear behavior. 3.2.3 Twolevel Correlation Function The numerical calculations of the cluster function are performed based on Y(u, v) [K(u, v)]2 (326) : = I.U . . i I I = 0 .9 ................ X=0.7  x=0.5  0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Figure 34. Eigenvalue density (x = 0 104) for Aensembles for A (7 = 0.50), 0.5 (7 = 0.25) as well as 1 (7 = 0.75). 0.9 (7 = 0.75), 0.7 0 5 10 15 20 25 30 35 40 Figure 35. Eigenvalue density (x = 0.50), 0.5 (7 = 0.25) = 0 40) for Aensembles for A = 0.9 (' = 0.75), 0.7 (7 as well as 1 (7 = 0.75) I I I I I I I 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Figure 36. Eigenvalue density (x = 0 104) for Aensembles for A (7 = 2.00), 1.5 (7 = 4.00) as well as 1 (7 = 0.75). 1.1 (7 = 0.75), 1.3 0 5 10 15 20 x Figure 37. Eigenvalue density (x = 2.00), 1.5 (7 = 4.00) 25 30 35 40 = 0 40) for Aensembles for A = 1.1 (7 = 0.75), 1.3 (7 as well as 1 (7 = 0.75). X = 1 1 ............... II X=1.3 X=1.5  0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 In[In(x)] Figure 38. Eigenvalue density for Aensembles for A (7 = 0.25) as well as 1 (7 = 0.75). 0.9 (7 = 0.75), 0.7 (7 = 0.50), 0.5 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 In[In(x)] Figure 39. Eigenvalue density for (7 = 4.00) as well as 1 Aensembles for A (7 = 0.75). = 1.1 (7 = 0.75), 1.3 (7 = 2.00), 1.5 with the unfolding map, e.g., u(x) dxp(x). (327) 3.2.3.1 Observation of Normal/Anomalous Structure of Twolevel Kernel It is well known that for A =1, the two level cluster function has both normal and anomalous component. The normal part of the twolevel cluster function for the critical ensembles for uv > 0 and lu v << u is given, in the 7 < 2r2 limit, by the sinh kernel [ 7 sin[7(u v)] (328) Y, (u, v)  s[ ))] (328) ) 27r sinh[ (u v)] The anomalous part of the cluster function, so called "ghost correlation peak" [54] for uv < 0 given by [7 sin[7(u v)] 2 Y (u, v)= [ csh[( )] 2 (329) The presence of the anomalous component occurring due to long range correlation is required by the normalization sum rule [57] 1 = du [Y,(u, u')+ YY(u, u')]. (330) The deficiency of the sum rule S1 duY(u,u')= duY (uu') (331) 0C J 0C is related to certain characteristics of the critical statistics [8, 3642]: i) the level compressibility in the number variance and ii) the multifractality of eigenvectors. In particular, d ((L)) d D, X= = (332) d(L) 2d where the fractal dimensionality Dp determines the scaling behavior of the moments of the inverse participation ratio via (< ddx (x) 2p) O LD(p1) (333) (7 It turns out that for A 4 1, Aensembles also possess such normal/anomalous structure. Figures 310 and 311 show the comparison between the normal part of cluster function for A 4 1 and that for A = 1 as well as that for the Gaussian. While the nodes of the cluster function remains the same (occurring at the integer value on the horizontal axis) for all A values as well as the Gaussian case, the peak height and position between the nodes show an interesting behavior in the change of A; as the A value is decreased for a given 7 value (7 = 0.5 in Fig. 310 and 7 = 2 in Fig. 311 ), the peak height and position gradually reduces and shifts toward the node on the left. The corresponding behavior is observed in the anomalous part as well. The figures 312 and 313 show the numerical evaluation of 1 Ya(u, v) (u > 0 and v < 0) for a symmetric range around v = u for varying A values for a fixed 7 (7 = 0.5 in Fig. 312 and 7 = 2 in Fig. 313). As the figure shows clearly, the magnitude of the ghost peak depends on A in a significant way; for A < 1, the peak is more pronounced than that for A = 1 and for A > 1 it is the opposite. The observation that such long range correlation leading to the ghost peak is preserved for all A 4 1 seems to suggest that such features, to some degree, are common to all logarithmic confinement potentials. In other words, once the critical ensembles break the U(N) symmetry of the Gaussian ensembles with the introduction of the parameter q, the A ensembles remain in this broken symmetry family. The fact that as A becomes large, the ghost peak shrinks seem to imply that the U(N) symmetry might become fully restored in the limit of A oc. This expectation seems consistent with the asymptotic behavior of twolevel correlation in the limit of A oo that will be shown later. The Aensembles are all "critical" in the sense that the twolevel kernel violates the sum rule that can be associated with the characteristics of the critical statistics such as the level compressibility and the multifractality. In particular, the fact that the violation of the sum rule is controlled by the parameter A, e.g., 0 < x(A) < 1 is intriguing. As mentioned above, A seems related to the degree of the U(N) symmetry 005 I Gausslan  X=I 0  0045 9 09 . 045' *X=08  X=07  004 0 035 003 ^ 0025 002 0015 / 001 i 0005 0 1 15 2 25 3 35 4 45 5 s Figure 310. Normal component of the cluster function for Aensembles for A = 0.9, 0.7, 0.5 as well as 1.0 (7 = 0.50). breaking and thus, indicative of the nontrivial character of the eigenvector correlations, namely the multifractal dimensionality as can be seen immediately from the Eq. 332 and the Adependent sumrule deficiency. In this regard, the study of the dimensional dependence of the critical statistics will be important to further understand the role of the parameter A since the multifractal dimension of the eigenvector correlations at the critical states is dependent on the spacial dimension [63]. 3.2.3.2 Twolevel Kernel of Aensembles It is known from rigorous results that for the softconfinment potential of the critical ensembles, the translationalinvariance is broken and the density does not depend on N. Thus the unfolding procedure is nontrivial [54, 64, 65]. In fact, the deformation from the Gaussian universality, the 'sine kernel', is the result of the nontrivial unfolding [54, 64, 65] In the semiclassical regime (7 < 27 for the critical ensembles), the kernel for an arbitrary weight function can be written as [6567] sin(w(u v)) x(u, ) () () (334) x(u) y(v) 005 I Gausslan  0045 =2   i,. = 1 0 004 0035 003 S0025 002 0015  001 , 0005 . 0 1 15 2 25 3 35 4 45 5 s Figure 311. Normal component of the cluster function for Aensembles for A = 1.1, 1.2, 1.3 as well as 1.0 ( = 1.50). I ' I lI 0.97  0.96  0.95 \ / 0.94 X=1.0 0.93 k=0.9 ................ X=0.7  k=0.7  0.92 I I I1 9 8.5 8 7.5 7 6.5 6 5.5 5 v Figure 312. Anomalous component of the cluster function for Aensembles for A =0.9, 0.8, 07 as well as 1.0 (7 = 0.50). 0.99 / \ . 0.98 0.97 U 0.96 0.95 0.94 X=1.0 0.93 =1. 1 ......... X=1.3  k=1.3  0.92 I I I I 8 7.5 7 6.5 6 5.5 5 4.5 4 v Figure 313. Anomalous component of the cluster function for Aensembles for A =1.1, 1.2, 1.3 as well as 1.0 (7 = 1.50). Thus we expect that in the same limit the difference between the Aensembles and the critical ensembles also arises from the difference in unfolding. To implement this, we define the unfolding variable for the critical ensembles in the following form, X c u' u U dt = Inx/xo (335) In the similar manner, we can define the unfolding variable for Aensembles for A / 1, u'= u U dt = ([In x]l [In xo]A) (336) t where u0 = jox p(x)dx and xo >> A, the cut off of the density that arises as a result of the regularization close to the origin. The constant c oc p(x = 0). Note that p(x = 0) is a function of the parameters that do not depend on N. For the critical ensembles, p(0) = 1 and for the Aensembles, p(O) = f(A, 7) where f is some function of A and 7. Rewriting the original variable x in terms of the unfolding variable, we obtain for both ensembles x = exp [7*(u + a)l' ] (337) where 7* (')1/ and the a [Inxo]^. In the limit A = 1, it reduces to x = xoe . (338) Thus for the critical ensembles we find the expected form of the kernel by using Eq. 334 1 x y sinh[ (u v)]. 2c (339) For A # 1, the same procedure results in x y sinh[ [(u' + a) (v' a)' ]]. 2 (340) Thus if we redefine the unfolding variables by u u = u' + a and u v = v'+ a, we obtain a suggested form of the twolevel kernel for the Aensembles r sin[7(D v)] Kn(,, 2 ) = sin )] (341) 2x sinh[ (ul/^ ,1/,)]' where F is introduced to ensure that the kernel satisfy the condition o(u) = lim, (u =) = 1. To identify the necessary form of F, we consider the kernel in the limit of v u lim K (u, v) v> F v 7* l/1A 1/A from which we obtain F = F(u, v) 7y*g(D, v) (343) Here, we introduce a function g(b, v) D /A I 1/X g(u, v) uv (344) Then, we finally obtain the proper form of the regular component of the twolevel kernel (uv > 0) for the Aensembles as (345) (342) S(uD, sin[7r(u i)] Sn( 2  27 sinh[l ( t)] while the anomalous part of the kernel (Du < 0) for the Aensembles can be written as a(, v) = (, ) sin[( (346) 27 cosh[ (u 9v)] We note that this kernel resembles the usual sinh kernel of the critical ensembles with more general argument. For A = 1, F(u, v) 7* = 7. Thus it reduces to the sinh kernel. However, in general for A / 1, the F(u, v) is a non trivial function of u and v. For example, F(u, ) = *(D + ) for A = 0.5 (347) and (u, ) = for A = 2.0 (348) VIU+ vv It is obvious that the form of the function is not translationally invariant. However, if we choose v to be a fixed value, i.e, v = v uo a = a, which is the same as choosing v = uo, then = u uo + a = u v + a s +a. In this way, the function F(u, v) and g(u, v) can be written in terms of a difference variable s u v = alone with a constant v = a that serves as a fixed reference point. F(s + a, a) F(s, a) (s = a) (349) Then F(s, a) sin(ws) K The figures 314, 315 shows the fitting results with the kernel given in Eq. 350 for different A values. They show a fit with the numerically obtained twolevel kernels with fit values 7* 7 = 0.5 and a 2 for A < 1 (Fig. 314) and 7* 7 = 1.5 and a 2 for A > 1. (Fig. 315) Finding the exact dependence of a and 7* on the parameter A and 7 requires to know details of the eigenvalue density p(x) in the vicinity of origin and the cutoff. This bears no importance in the discussion of the universal feature of the ensembles because p(x) in the vicinity of the origin is simply a reflection of how one chooses V(x) to 005 Gausslan  X= 1 0  0045 / X =O 9 (fit) X=O 8 (fit) X=O 7 (fit) 004 X1= 9  )=O 8  S=O 7   0035 003 S0025 002 0015 001 / \ 0005 \ , 1 15 2 25 3 35 4 45 s Figure 314. Fitting results for normal component of the cluster function for Aensembles for A = 0.9, 0.7, 0.5 as well as 1.0 (y = 0.50). be regularized near the origin. We also show that in the similar range of the parameters, the anomalous components of the kernel fits well with the proposed form of the kernel Eq. 346 which are shown in the figures 316 and 317. 3.2.3.3 Universality of Aensembles In the following, we will discuss novel asymptotic behavior of the twolevel kernel of the Aensembles by further examining Eq. 350. First, we note that F(s, a) oc sl for s > a (351) leads to K,(s, a) oc s 1'eS'/ for s > a. (352) Looking at the Adependence, we observe that Kn(s,a) oc es for A O oc e for A = 1 o s1 for A oo. (353) 0 045 / =1 J (Tit) !i\,, 1= 1 2 (fit) ', 1=1 1 (fit) 004 1=1 3 A=I 2  =X 1  0 035 003  P 0025 002 ' 0015 \ 0 01  0005 ... 1 15 2 25 3 35 4 45 5 s Figure 315. Fitting results for normal component of the cluster function for Aensembles for A = 1.1, 1.2, 1.3 as well as 1.0 ( = 1.50). 0.99  0.98  0.97  0.96  0.95 S 0.94 X=1.0  0.93 k =0.9 (fit) X=0.8 (fit) 0.92 =0.7 (fit) X=0.9 ............... 0.91 X=0.8  X=0.7 . 0.9 I I I 1 9 8.5 8 7.5 7 6.5 6 5.5 5 v Figure 316. Fitting results for anomalous component of the cluster function for Aensembles for A =0.9, 0.8, 07 as well as 1.0 (7 = 0.50). 0.99/ \C / 0.98 0.97 .96 S 0.95 0.94 =1.0  X=1.1 (fit) 0.93 l=1.2 (fit) k=1.3 (fit) = 1.1 .............. 0.92 =1.2  k=1.3  0.91 I I I 8 7.5 7 6.5 6 5.5 5 4.5 4 v Figure 317. Fitting results for anomalous component of the cluster function for Aensembles for A =1.1, 1.2, 1.3 as well as 1.0 (7 = 1.50). Thus in the limit A  oo, we get back the wellknown asymptotic sine kernel decay. This is consistent with the fact that the magnitude of the ghost correlation peaks become smaller as A increases, presumably disappearing in the Gaussianensemble limit of very large A. For A = 1, the kernel shows the expected exponential decay of the critical ensembles. For A + 0 limit, the asymptotic tail is given by the infinitely fast exponential decay, leading to the uncorrelated Poissonlike behavior. In general, for all values of A, the large s behavior is governed by an exponential decay of es"' In the same token, we can study the twolevel correlation function or densitydensity correlation function defined by R(x, y) (p(x)p(y)) 1 (354) (p(x))(p(y)) 6(x y) (p(x)p(y)) = Y+ 1 (355) (p(x)) (p(x)) (p(y)) xy In the unfolding scale p(u) = p(x)dx where p(u) = 1, we can rewrite R(u, v) = 6(u v) (p(u)p(v)),}v 1 = 6(u v) Y(u, v) (356) In terms of the difference variable s = u v, we define the twolevel correlation function for the Aensembles as R (s) 6(s) YA(s) (357) For s < a, RX(s) ~ s2 for all values of A, which is an expected feature due to the unitary symmetry of the ensembles. For large s > a, the results are simply obtained from Eq. 353 R'(s, a) oc es for A 0 (358) oc e forA= 1 (359) oc s2 for A oo (360) We can see that the asymptotic behavior of the twolevel correlation function interpolates that of Gaussian ensembles for A oo limit, that of critical ensembles A = 1 and the free Levy matrices for A  0 limit. 3.2.4 Number Variance As an example of the level statistics obtained from the proposed form of the twolevel kernel Eq. 350, we show the evaluation of the number variance E(L) within a range L shown in Figure 318. It clearly shows that the number variance is linear in L the for all values of A with a fixed 7 = 0.5. As A become smaller, the slope of the number variance increases. This is consistent with the fact that deficiency of sum rule rule X(A) increases as A decreases since the slope of the number variance is directly related to the deficiency of the sum rule by X(A) = dZ)) dKL) 3.3 Discussion In this work, we further study an invariantclass of random matrix ensembles characterized by logarithmic softconfinement potentials introduced in Ref. ?? which we refer to as the Aensembles. As a first step, we carefully reinvestigate the spectral density of the Aensembles and show that the spectral density is given more 18 16 14 " 12 . 1 2  08         "" ~ 06 =0 5 ...... 5 :=0 8  Critical 04 X 1 5 . X=2 0  Gaussian  02 I I 0 5 10 15 20 25 30 35 40 L Figure 318. Number variance for Aensembles for A =0.5, 0.8, 1,0, 1.5, 2.0 (7 = 0.5). accurately by a powerlaw of the form p(x) o [In x]A'/x. This result is suggested by the meanfiled approach and it can be checked by the normalization condition of the spectral density, the edge of which is determined by the coefficient Rn of the Ageneralization of qpolynomials. Second, we show that the twolevel kernel of the Aensembles has normal/anomalous structure, which is characteristic of the critical ensembles. The anomalous component arising due to the sum rule violation is dependent on the parameter A for a fixed value of 7 ~ 0(1); As the value of the A decreases, the deficiency of the sum rule becomes larger. Third, we identify the normal and anomalous components of the twolevel kernel in the semiclassical regime, which are given by Eq. 345 and Eq. 346 that reduce to those of the critical ensembles for A = 1. Further, we show that the twolevel kernel of the Aensembles exhibit novel universal asymptotic behavior, shown by Eqs. 353, which includes the Gaussian ensembles (A oo limit), the critical ensembles (A = 1) as well as the free Levy matrices (A + 0 limit). In particular, the large s behavior of the twolevel kernel is governed by exp[s1/], which is a novel feature of the Aensembles. It is expected that the asymptotic tail of the spacing distribution is also given by a similar exponential form. Lastly, we show that the number variance is linear in L for large L. The slope of number variance or the level compressibility is dependent on the parameter A. As A decreases, the slope increases, which is consistent with that fact that anomaly or the deficiency of the sum rule increases as A decrease. These results seem to imply that the Aensembles in general are relevant to the description of the critical states of the localizationdelocalization transition in the disordered systems. Since the critical level statistics are universal, depending only on the critical exponent and the dimensionality of space for a given symmetry class, it is conceivable that the parameter A can be associated with these parameters. In particular, the study of the dimensional dependence of the critical statistics is interesting in this respect. At the same time, it is also interesting to see if Aensembles are also applicable in the study of quantum chaos. We note that recently the critical statistics has been found relevant in some cases of quantum chaos as well [68, 69]. It turns out that the two level kernel of chaotic systems with logarithmic singularity [70] have the exact same form as that of the critical ensembles. CHAPTER 4 CONCLUSION In this work, we study invariantclass of random matrix ensembles characterized by the asymptotic logarithmic softconfinement potentials, named Aensembles. The suggestion is inspired by the existing RM models such as the critical ensembles (A=1), the free levy matrices (A + 0 limit) and the Gaussian ensembles (A oo limit) in an effort to investigate the novel universality associated with the fattail RMEs within the rotationally invariant RMT framework. The following is the summary of the main findings: * The polynomials that are orthogonal with respect to the arbitrary power of the asymptotic logarithmic potential belong to a novel class of orthogonal polynomial system, named "Ageneralization of qHermite polynomials". The onelevel correlation (the spectral density) of the Aensembles are given by a powerlaw form. The unfolded twolevel correlation function shows the normal/anomalous structure, which is the characteristic of the critical ensembles. In particular, the sum rule violation is controlled by the parameter A for a fixed value of 7 0(1). As the value of A decreases, the sum rule violation becomes greater. The asymptotic tail of the densitydensity correlations show a novel form shown in Eq. 358; for 1 < A < oo, the asymptotic tail interpolates between the critical ensembles (A = 1), the Gaussian ensembles (A oo), and the uncorrelated Poissonlike statistics (A + 0). These results seem to have interesting applications/implications: * It is of immediate interest whether these results are relevant in the context of the Anderson transition in disordered systems as well as in a broader context of the localizationdelocation problems. It would also be interesting to see if the these results are applicable in the mixed states of the quantum chaos systems where the spectral properties are intermediate between WD and Poisson statistics. These results strongly impliy that the nontrivial Ndependence of twolevel kernel of the fattail RMEs is a generic feature within the framework of the rotationally invariant RMT. In the A  0 limit of the Aensemble, 7 is required to be a Ndependent parameter to have the probability measure to be normalizable, which is the case for the freelevy matrices. Thus, the Ndependence of the twolevel kernel in this limit can be understood as a consequence of the presence of the Ndependent parameter in the model ensemble, which cannot be simply scaled out. For the future, we further need to study the level statistics of the Aensembles such as the spacing distribution, the number variance, the spectral form factors, the inverse participation ratio etc. both analytically and numerically. These will allow much broader applications of our results. In addition, it is also important to explore if there exist a novel universality of random matrix ensembles associated with the fattail distribution in the framework of the Wignerclass (noninvriant) random matrix ensembles. APPENDIX A CENTRAL LIMIT THEOREM It is understood that the universality of the Gaussian ensembles has some connection to the central limit theorem1 The fact that there are two kinds of stable probability distribution, e.g. the Gaussian/Levy basin according to the central limit theorem2 inspires us that there would exist the counterpart structure in RMT, namely * Gaussian distribution Gaussian RMT * Levy distribution Levy RMT Yet, this conjecture is not proven although there have been several attempts. Here, we will review the content of the central limit theorem. The central limit theorem states that the distribution of the large sum of independent identically distributed (i.i.d) random numbers reach to certain limiting form of probability distributions. Suppose we consider a sum X of a large number (N) of i.i.d random numbers, x, drown from a certain probability distribution, X = +X2 +x3 ... +XN. (A1) The sum X can be considered as the position of a particle undergoing a Brownian motion, each step size of which is given by a random number xi. After the large sum is done or the large number of steps are taken, it is proven that the probability distribution of the sum X, or analogously that of the position of the particle under the Brownian motion converges to a limiting distribution, namely, the Gaussian distribution as long as 1 In fact, it can be proven that the eigenvalue distribution of large random matrices becomes the well knownsemi circle using the central limit theorem [71]. 2 To be precise, it should be referred to as "generalized central limit theorem". the each individual random variable xi are sampled from the distributions with a finite variance regardless of other details. 3 For a short proof, we define the characteristic function P(k) of a probability distribution as P(k) = dx ekx P(x) (A2) After the large sum X = C" x,, the probability distribution of P(X) is given as the inverse fourier transform of N times convolutions of P(k). P(X) = eik [(k)] (A3) 27dk P(k) has the cumulant expansions around k = 0 in the following way, In (k)n (ik) (A4) n=0 where ci denotes the cumulants of distributions. The first few of them are named; cl is mean, c2 is the variance, c3 skewness and c4 kurtosis etc. Then, we can write P(X) as P(X) = eNf(k,X) (A5) S27dk where f(k, X) is given by X n f (k, X) = ikn (ik)n (A6) N n n=0 Using the method of steepest decent, we know that in the large N limit, the dominant contribution of f(k, X) comes from the maximum of f(k, X), which is at k = 0. Thus, in the rescaled variables x = Nc and w = c2Nk, we obtain the expansion of P(x) in 3 The existence of the higher order moments determine how fast it converges to the limiting form. powers of N1/2. It is shown that e"2 h3(x) h4 (X) P(x) ~ [1 + ...] (A7) where h,(x) can be written in terms of the hermite polynomials of order n. This approximation breaks down when h3(x)/vN becomes 0(1). It means that the width of the Gaussian distribution approximation scales x O(N2/3). In case the variance of the probability distribution does not exist, the central region collapses faster than N log N and the probability distribution of large sum converges to a different limiting function, known as Levy distribution. In particular, in the large x limit, the distribution shows a powerlaw tail of the form, A P(x 00) ~ xl; 0 < a < 2 (A8) X1 a APPENDIX B NDEPENDENCE PROBLEM IN SPREAD FUNCTION APPROACH The "Spread function approach" is one of the generalization strategies of the Gaussian unitary ensemble [24]. The underlying idea of this approach is to work with associated characteristic function, not directly probability distribution itself, to discover a novel random matrix ensemble. In this framework that the general npoint correlation function are given by Eq. 115. Here we will show how the nontrivial N dependence arises when the spectral density designed to give a fattail distribution. As was observed in our numerical study 1 we need to search for the novel universality in x oo and N oo limit where the fattail spectral density is expected to occur. We find [72] that there is an asymptotic expression for harmonic oscillator function V(x) in the limit where x 0 o and N o so u = finite, e.g., x = 2Ncos . 21/4 N1/4 \ N 1\ 3r 11 \VN(x) 2 ""n sin + ')(sin 2 2) + + )} (B1) \vin (2 4 4 N Using the ChristoffelDarboux formula, we can write down the twolevel kernel. 1 (x y)K(x, y) = H(x, y) (B2) 27r sin 0sin " where x = v/2Ncos y = v/2cos and u(0) = sin(20) 20 and. H(x,y) = 2sin (u() u()) sin (u() u()) 2 cos N(u() + u())] sin [(u() u()) 1 We investigated Levy matrices discussed in Chapter 1 numerically. It turns out that the density oN ~ N for large x. So it is expected that in a double scaling limit, e.g., x o and N oo such that u = finite, the density can be unfolded (u) = 1 and a novel universality may show up in terms of this variable u. From the kernel, density of states reads directly as oN(x) = limyx K1(x, y). (B3) For the density Eq. B3, it turns out that choosing the spread function f(b) (b(~ +l)ep/b give rise to a fattail density, e.g., ,N(x) ~ F 2 dM2e sin +1/cos  aJxNV 2N(8N) \easi cosf1 12V'e _2] F (A) \x J [ J L J (B4) where = finite in the limit x oo, N o and b ~ N and a So if one V8Nb X2 choose A = and /N, then ON(X) goes as 7. In order to study unfolded correlation function, we define a scaling variable u = N such that (u) = 1. In particular, the unfolding variables p aN and = aN where a is some constant. Then the unfolded cluster function some constant. Then the unfolded cluster function Y2(P, ) lim T2 (xr, x,). Ntoo dp dx Since T2 = T2 6T2, similarly we define Y2 = Y2 _ Y2 so that yO(,Y) lidx dy dbf(b)G Noocdp dX J 4b and Y dx dy dbf(b) G G) 6Y2(p,X) Ilm  (x1)_(x2) 1 No dp dX Jo 4b For further calculation, we begin with 6Y2. First, we rewrite o1(x) in terms of the unfolding variable p, (B5) (B6) (B7) G( ) G= aNg) sin co1 a28b p ' )2bN( ( 2COs 8b p a2 n a2N 1 8b p 8b p2J Plugging this into eq. B7, we get SY2(PN a2N2 / dbf(b) a aN G aN 6Y2(p, X) lim b 2 2aI /  N/ p2X Jo 4b p 247 (B8) o(x) ~ Msin(O sin dOcos 0) (B9) After substituting f(b) and making a change of variable, 2= 2 dependence can be taken out from the integral. Thus we have, 8 a2N2 A p (B8) 6Y2(p, X) lim 22 /) ) (B10) Noo a2 p2X2 F(A) aN where (; p, = d e 2 in [ os (1 2)] sin [ cos1 E 1 2 )] (B11) In the limit N oo, ~ N, with other variables order of 1, the argument of exponential in Eq. B 1 can be made very small. So the leading order of the integral Eq. B 1 goes as 1. With this in mind, putting A =1 into Eq. 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GarciaGarcia, Phys. Rev. E 73, 026213 (2006). [70] A. M. GarciaGarcia and J. Wang, Phys. Rev. Lett. 94, 244102 (2005). [71] J. Bouchaud and M. Potters, Theory of financial risk and derivative pricing: from statistical physics to risk management (Cambridge University Press, 2003). [72] M. E. H. Ismail, private communication (2006). BIOGRAPHICAL SKETCH Jinmyung Choi was born in Pusan, South Korea in November 1973. He received his Ph.D. from University of Florida in the summer of 2010. PAGE 1 RANDOMMATRIXENSEMBLESWITHSOFTCONFINEMENTPOTENTIALByJINMYUNGCHOIADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2010 PAGE 2 c2010JinmyungChoi 2 PAGE 3 Tomyfamily 3 PAGE 4 ACKNOWLEDGMENTS Firstofall,Iwouldliketothankmyadvisor,KhandkerMuttalibforallthehelp.Iwouldalsoliketothankhimforallthevaluablediscussions,whichledmetounderstandvariousaspectsofrandommatrixtheoryandtodevelopmyinterestsaroundthesubject.Second,Iwouldliketothankmyfamilyandfriendsfortheirconstantloveandencouragement.Andlastly,IwouldliketothankDepartmentofPhysicsatUniversityofFloridaforofferingmethevaluableteachingexperiences. 4 PAGE 5 TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFFIGURES ..................................... 7 ABSTRACT ......................................... 9 CHAPTER 1MOTIVATION ..................................... 11 1.1RandomMatrixTheoryandComplexSystems ............... 11 1.2ComplexSystemswithPowerlawDistributions ............... 12 1.3FattailRMModels ............................... 15 1.3.1LevyMatrices .............................. 15 1.3.2FreeLevyMatrices ........................... 16 1.3.3NonextensiveqEnsembles ...................... 18 1.3.4EnsemblesBasedonSuperpositionsofClassicalorGaussianEnsembles ............................... 19 2RANDOMMATRIXTHEORY ............................ 22 2.1RandomMatrixEnsembles .......................... 22 2.2OrthogonalPolynomialMethod ........................ 24 2.2.1JointProbabilityDistributionFunction(JPDF) ............ 25 2.2.2DeterminantFormofJPDFandKernel ................ 26 2.2.3CorrelationFunctionsandClusterFunctions ............. 28 2.2.4Unfolding ................................ 29 2.3GaussianEnsembles ............................. 30 2.3.1SemiCirclelaw ............................. 30 2.3.2SineKernel ............................... 31 2.3.3GapProbabilityandSpacingDistribution ............... 32 2.3.4NumberVariance ............................ 33 2.4CriticalEnsembles ............................... 33 2.4.1AndersonTransitionandCriticalEnsembles ............. 33 2.4.2CriticalStatistics ............................ 34 2.5CoulombGasAnalogy ............................. 35 3ENSEMBLES .................................... 38 3.1Method ..................................... 39 3.2Results ..................................... 42 3.2.1generalizationofqpolynomials ................... 42 3.2.2EigenvalueDensity ........................... 44 3.2.3TwolevelCorrelationFunction ..................... 46 5 PAGE 6 3.2.3.1ObservationofNormal/AnomalousStructureofTwolevelKernel ............................. 50 3.2.3.2TwolevelKernelofensembles .............. 52 3.2.3.3Universalityofensembles ................. 57 3.2.4NumberVariance ............................ 60 3.3Discussion ................................... 60 4CONCLUSION .................................... 63 APPENDIX ACENTRALLIMITTHEOREM ............................ 65 BNDEPENDENCEPROBLEMINSPREADFUNCTIONAPPROACH ...... 68 REFERENCES ....................................... 71 BIOGRAPHICALSKETCH ................................ 75 6 PAGE 7 LISTOFFIGURES Figure page 31LogRnasafunctionofnfordifferentvaluesof.Solidlinecorrespondstothecriticalensemble=1. ............................. 43 32Theexponent()asafunctionoffortwodifferentvaluesofln(1=q).=1correspondstotheqpolynomialsdescribingthecriticalensembles. ... 44 33Densityofeigenvaluesfordifferentvaluesof. .................. 45 34Eigenvaluedensity(x=0)]TJ /F6 11.955 Tf 11.02 0 Td[(104)forensemblesfor=0.9(=0.75),0.7(=0.50),0.5(=0.25)aswellas1(=0.75). ................... 47 35Eigenvaluedensity(x=0)]TJ /F6 11.955 Tf 12.08 0 Td[(40)forensemblesfor=0.9(=0.75),0.7(=0.50),0.5(=0.25)aswellas1(=0.75) ................... 47 36Eigenvaluedensity(x=0)]TJ /F6 11.955 Tf 11.02 0 Td[(104)forensemblesfor=1.1(=0.75),1.3(=2.00),1.5(=4.00)aswellas1(=0.75). ................... 48 37Eigenvaluedensity(x=0)]TJ /F6 11.955 Tf 12.08 0 Td[(40)forensemblesfor=1.1(=0.75),1.3(=2.00),1.5(=4.00)aswellas1(=0.75). ................... 48 38Eigenvaluedensityforensemblesfor=0.9(=0.75),0.7(=0.50),0.5(=0.25)aswellas1(=0.75). .......................... 49 39Eigenvaluedensityforensemblesfor=1.1(=0.75),1.3(=2.00),1.5(=4.00)aswellas1(=0.75). .......................... 49 310Normalcomponentoftheclusterfunctionforensemblesfor=0.9,0.7,0.5aswellas1.0(=0.50). ............................ 52 311Normalcomponentoftheclusterfunctionforensemblesfor=1.1,1.2,1.3aswellas1.0(=1.50). ............................ 53 312Anomalouscomponentoftheclusterfunctionforensemblesfor=0.9,0.8,07aswellas1.0(=0.50). .......................... 53 313Anomalouscomponentoftheclusterfunctionforensemblesfor=1.1,1.2,1.3aswellas1.0(=1.50). .......................... 54 314Fittingresultsfornormalcomponentoftheclusterfunctionforensemblesfor=0.9,0.7,0.5aswellas1.0(=0.50). ................... 57 315Fittingresultsfornormalcomponentoftheclusterfunctionforensemblesfor=1.1,1.2,1.3aswellas1.0(=1.50). ................... 58 316Fittingresultsforanomalouscomponentoftheclusterfunctionforensemblesfor=0.9,0.8,07aswellas1.0(=0.50). .................... 58 7 PAGE 8 317Fittingresultsforanomalouscomponentoftheclusterfunctionforensemblesfor=1.1,1.2,1.3aswellas1.0(=1.50). ................... 59 318Numbervarianceforensemblesfor=0.5,0.8,1,0,1.5,2.0(=0.5). .... 61 8 PAGE 9 AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyRANDOMMATRIXENSEMBLESWITHSOFTCONFINEMENTPOTENTIALByJinmyungChoiAugust2010Chair:KhandkerMuttalibMajor:PhysicsInthiswork,westudyinvariantclassofrandommatrixensemblescharacterizedbytheasymptoticlogarithmicsoftconnementpotentialV(H)[lnH](1+)(>0),named\ensembles.Thesuggestionisinspiredbytheexistingrandommatrixmodelssuchasthecriticalensembles(=1),thefreeLevymatrices(!0limit)andtheGaussianensembles(!1limit)inanefforttoinvestigatethenoveluniversalityassociatedwiththefattailrandommatrixensemblesaswellasthelogarithmicsoftconnementpotentialwithintheframeworkofrotationallyinvariantrandommatrixtheory.Firstofall,weshowthattheorthogonalpolynomialswithrespecttotheweightfunctionexp[)]TJ /F6 11.955 Tf 9.3 0 Td[((lnx)1+]belongtoanovelorthogonalpolynomialsystem,namedgeneralizationofqpolynomials.Second,weshowthatbasedonnumericalconstructionofthegeneralizationofqpolynomials,wecanstudytheonelevelandthetwolevelcorrelationfunctionsaswellasthelevelstatisticsoftheensembles.Third,weshowthattheonelevelcorrelation(eigenvaluedensity)hasapowerlawform(x)/[lnx])]TJ /F7 7.97 Tf 6.58 0 Td[(1=xandtheunfoldedtwolevelcorrelationfunctionpossessesthenormal/anomalousstructure,characteristicofthecriticalensembles.Wefurthershowthattheanomalouspart,socalledghostcorrelationpeakiscontrolledbytheparameter;decreasingincreasestheanomaly.Third,wealsoidentifythetwolevelkerneloftheensemblesinthesemiclassicalregime,whichcanbewritteninasinhkernelformwithmoregeneralargumentthatreducestothatofthecritical 9 PAGE 10 ensemblesfor=1.Forth,weshowthatthenumbervarianceislinearinLforallandtheslope(thelevelcompressibility)isincreasingasdecreases,whichisconsistentwiththedependenceofsumruleviolation0<()<1.Finally,wewilldiscussthenoveluniversalityoftheensembles,whichinterpolatestheGaussianensembles(!1limit),thecriticalensembles(=1),thefreeLevymatrices(!0limit). 10 PAGE 11 CHAPTER1MOTIVATION 1.1RandomMatrixTheoryandComplexSystemsRandommatrixtheory(RMT)dealswitheigenvalueandeigenvectorcorrelationsofrandommatrixensemblesdrawninastochasticmanner.TheearlierapplicationofRMT[ 1 2 ]inphysicsdatesbacktothe1950swhenWignerintroducedRMTtoexplaintheuctuationpropertiesofheavynucleienergyspectrum.Inaslownuclearreaction,incidentnucleoninteractswiththeconstituentsoftargetnucleusinacomplicatedmannersuchthatextraenergycarriedinbytheanincidentnucleonissharednontriviallywiththenucleonsofthetargetnucleuswhileformingtheexcitedstates.However,theuctuationpropertiesoftheexcitationenergylevelsofthecompoundnucleusarenotverywellunderstood[ 3 ].Toexplainthestatisticalnatureoftheenergyspectrumofthecompoundnucleus,WignersuggestedtoconsiderensemblesofHamiltonianswhoseentriesarerandomlydrawnfromtheGaussiandistribution.ItturnsoutthatthestatisticalpropertiesoftheeigenvaluecorrelationsofsuchrandomHamiltonianensembles,e.g.theGaussianensembles,showedagoodagreementwiththestatisticalbehaviorsoftheeigenvaluespectrumobtainedfromvariousheavynucleireactionexperiments[ 3 4 ].TheunderlyingideaofRMTiswellexplainedbyDyson[ 5 ]:Whatisrequiredisanewkindofstatisticalmechanics,inwhichwerenounceexactknowledgenotofthestateofthesystembutofthesystemitself.Wepictureacomplexnucleusasablackboxinwhichalargenumberofparticlesareinteractingaccordingtounknownlaws.AsintheorthodoxstatisticalmechanicsweshallconsideranensembleofHamiltonians,eachofwhichcoulddescribeadifferentnucleus.Thereisastronglogicalexpectation,thoughnorigorousmathematicalproof,thatanensembleaveragewillcorrectlydescribethebehaviorofoneparticularsystemwhichisunderobservation.Theexpectationisstrong,becausethesystemmightbeoneofahuge 11 PAGE 12 varietyofsystems,andaveryfewofthemwilldeviatemuchfromaproperlychosenensembleaverage.Oneofthesurprisingaspectsofthisnewkindofstatisticalmechanicsisthatitsutilityisfarreachingbeyonditsorigination.Sinceitsintroductiontothemanybodycomplexnucleisystems,RMThasbeenappliedtoawidevarietyofsystemsindiverseareas,includingmanybodyatomsandmolecules,quantumchaos,mesoscopicdisorderedconductor,2Dquantumgravity,conformaleldtheory,chiralphasetransitionsaswellaszerosofRiemannzetafunction,scalefreenetworks,biologicalnetworks,communicationsystemsandnancialmarkets[ 6 18 ].ThisbroadrangeofapplicabilityofRMTinseeminglyunrelatedareashighlightstheuniversalfeaturesofthecorrelationsoftheeigenvaluesinRMT.WithintheclassicalorGaussianmodelpioneeredbyWigner,thesecorrelationsareknownastheWignerDyson(WD)statisticsoftheGaussianensembles,whicharequalitativelydifferentfromthestatisticalfeaturesofcompletelyuncorrelatedeigenvaluesgivenbythePoissonstatistics. 1.2ComplexSystemswithPowerlawDistributionsInthepastdecades,therehavebeengrowinginterestofRMTapplicationstogenericcomplexsystemsthatappearinthestudyofnatural,social,economicalandbiologicalsystems.Inthesesystems,themicroscopicinteractionsamongtheconstituentsaresocomplicatedthatthestatisticalhypothesismadeinRMTseemsrelevant.Namely,weareforcedtorenounceexactknowledgenotofthestateofthesystembutofthesystemitself.Thus,itappearstobeappropriatetoapplytheideaofRMTtounderstandcertainstatisticalfeaturesofthesesystems.OnesuchapplicationofRMTisfoundintheanalysisoftheempiricalcovariancematricesinamultivariatesettingsuchasnancialassetsandclimatesystems[ 11 12 19 ]basedonclassicalRMmodels.Intheseworks,theapplicationaimsatdecomposinggenuinecorrelations,presentintheeigenvaluesandeigenvectorsoftheempiricalcovariancematricesfrompurenoisecomponents,theRMTpredictions.Forexample, 12 PAGE 13 Refs.[ 11 12 ]studytheempiricalcrosscorrelationmatricesofthestockpricereturnsofS&P500andthelargest1000UScompaniesforcertainperiodtimesuchthattheelementsofsuchmatricesrepresentthecorrelationsamongthecompaniesinagivenperiod,e.g., Cijhri(t)rj(t)i.(1)Here<...>denotestimeaverageovertheperiodstudiedandri(t)representsanormalizedtimeseriesofstockpricereturnofacompanyidenedas ri(t)Ri(t))221(hRi(t)i i(1)wherethevolatilityip hRi(t)2i)222(hRi(t)ihRi(t)iandRi(t)representsthelogarithmofthepricechange, Ri(t)lnSi(t+t))]TJ /F3 11.955 Tf 11.96 0 Td[(lnSi(t)'Si(t+t))]TJ /F3 11.955 Tf 11.96 0 Td[(Si(t) Si(t).(1)Here,Si(t)isthetimeseriesofthestockprice.Thus,thecovariancematrixCijisbyconstructionarealsymmetricmatrix.ThestatisticalpropertiesofthecovariancematrixCijcanbecomparedtothewellknownresultsofappropriateclassicalRMmodels,e.g.Gaussianorthogonalensembles(GOE)orWishartmatrices(Laguerreensembles).ThestudyshowsthatmostoftheeigenvaluesexceptsomeoutliersarewellwithintheRMTexpectationsintermsoftheeigenvaluedensity,thenearestspacingdistribution,thenumbervarianceandtheinverseparticipationratio,implyingthattheempiricalcovariancematricesaredominatedbyrandomGaussiannoiseforthemostpart.Thesmallfractionoftheoutlierscarriessomemeaningfulinformationsusefulfortheriskmanagement.Thisexplanation,however,isnotfullysatisfactorybecausetheaprioriassumptionthatcrosscorrelationspresentinthestockpricereturnsaredominatedbytheGaussiannoiseisnotinaccordancewiththeobservationthatthedistributionofstockpricereturnsfollowsanasymptoticfattaildistribution,thesocalledinversecubiclaw[ 20 ].Tobe 13 PAGE 14 precise,itisshowninRef.[ 20 ]thatthecumulativedistributionofthestockpricereturnshasanasymptoticfattaildistribution,e.g.P(ri(t)r)r)]TJ /F8 7.97 Tf 6.59 0 Td[(with'3.Thus,theapplicationoftheclassicalGaussianRMmodelsrequiresreconsiderationinthisexampleandmoregenerally,inthecomplexsystemswheretheuctuationisbettercharacterizedbyfattailnoises.ThissituationmotivatessearchforsuitableRMmodelsthatcanincorporatethefattailnoises.Inthiscontext,manyattemptshavebeenmadetoconstructgeneralizedrandommatrixensemblesthatincorporatepowerlaworfattaildistributions[ 21 28 ].ThesignicanceofsuchgeneralizationbeyondtheGaussianensemblesismainlytwofold.First,therearenumerouscomplexsystemsthatexhibitfattailnoises,notably,nancialmarkets,earthquakes,scalefreenetworksetc.[ 29 32 ].Aspointedoutearlier,therelevanceoftheclassicalRMmodelsinthesesystemsseemsquestionable.ItisbecausetheclassicalorGaussianRMmodelsarebasedontheassumptionthatthesystemsarecharacterizedbytheGaussiannoise,whichisnotappropriateforsystemswithfattailnoisewheretheoccurrenceofextremeeventsarenotasrareasexpectedfromnormaldistributions.Second,itisconceivedthattheuniversalityofclassicalorGaussianRMmodelsiscloselylinkedtotheprevalenceofnormalorGaussiandistributionsinnature,aconsequenceofthecentrallimittheorem[ 21 ].Ithas,however,notbeenfullyinvestigatedifthereisacounterpartoftheGaussianensembleasimpliedbythestructureofthegeneralizedcentrallimittheorem[ 21 ],comparinge.g,theGaussianandtheLevybasins(refertoAPPENDIXAformoredetails).Sofar,fattaildistributionsinrandommatrixensembleshavebeencarefullyincorporatedinsomelimitedcasesandthecalculationofthecorrelationfunctionsoftheeigenvalueshavebeencarriedoutforcertainspecialcases[ 21 28 ].However,thequestionregardingtheuniversalityofthecorrelationsoftheeigenvaluesremainsunresolved.ForGaussianensembles,itisthewellknowntwolevelsinekernelthatestablishestheuniversalityofthecorrelationsintheproperlyscaledlargeN(matrix 14 PAGE 15 size)limitbutitisnotclearifthereexistsasimilaruniversaltwolevelkernelforthepowerlaworthefattailensemblesaswell. 1.3FattailRMModels 1.3.1LevyMatricesTheLevymatrices[ 21 ]areintroducedbyP.CizeauandJ.P.BouchaudinconsiderationofconstructingaRMmodelbasedontheLevyprobabilitydistribution.Inparticular,theyconsideredaNNrealsymmetricmatrixensembleofindependentindenticallydistributed(i.i.d.)randomnumbersofasymptoticLevydistributionsgivenbythefollowing: P(Hij)H0 jHijj1+;0<<2(1)whereHijdenotesthematrixelements.Byconstruction,Hij=HjiandH0issettobeN1=inordertoensurethatthetypicallargestelementofarowisO(1).Sincethematrixelementsaredrawnfromthedistributionswithdivergentmoments,thereisnodirectwaytoconstructananalyticalmodelwithinthestandardRMTtechniques.Thus,theydevelopedanovelanalyticaltechniquebasedonthecavitymethod[ 21 ]toevaluatetheeigenvaluedensityandtheinverseparticipationratioinparallelwithnumericalsimulations.Themainresultsoftheirworkaresummarizedinthefollowing: 1. TheeigenvaluedensityconvergestoalimitingformofdistributioninthelargeNlimitthathasasymptoticpowerlawdistribution.e.g.(x!1)1 x1+,whichhasthesamescalingpowerasthatofthematrixelements. 2. Fromcarefulexaminationsoftheinverseparticipationratio,itisshownthattheeigenstatesundergoanontriviallocalizationdelocalizationtransitionwithintheeigenvaluespectrumatacertaincriticalvalueofx=xcdependingontheparameter:for>1,thereexistsacriticalvaluexcthatdifferentiatetheextendedstates(x PAGE 16 3. Thenumericallyobtainednearestneighborspacingdistributionisfoundtobenonuniversal1,whichdependsonthelocationoftheeigenvaluespectrumconsidered. 1.3.2FreeLevyMatricesThefreeLevymatrices,introducedbyZ.Burdaetal.[ 22 ],areconstructedbasedonthetheoryoffreerandomvariable(FRV)thatofferstheprobabilitytheoryofnoncommutingvariablesasageneralizationoftheclassicalprobabilitytheory.ThecorrespondencebetweentheclassicalprobabilitytheoryandtheFRVtheoryismadeinRef.[ 33 ]: 1. ProbabilitydistributionP(x):Spectraldensity(x)=)]TJ /F3 11.955 Tf 9.3 0 Td[(ImG(x+i0)= 2. Characteristicfunction^P(k):GreenfunctionG(z) PAGE 17 realvaluedandareassociatedwiththeshift,theslope()andtheskewness()andtherange()ofthestabledistributions.NotethatR(z)isgivenwithoutanyparticularrealizationofmatrixensembles.Inordertondaparticularrealizationofarandommatrixensemblecharacterizedbytheprobabilitymeasureoftheform e)]TJ /F5 7.97 Tf 6.59 0 Td[(NTrV(M)dM(1)whereTristhetraceandV(M)isspecictotherealizationoftheensemblesandisrelatedtotheconningpotentialintheCoulombgasanalogyoftheclassicalRMmodelsdevelopedbyDyson[ 34 ],Z.Burdaetal.[ 22 ]rstcalculatedthespectraldensities3fromthestableRtransformsforsomeexactlysolvablecasesofparameters.TheyshowedthattheasymptoticformofspectraldensitiesdisplayspowerlawdistributionswiththeexponentswithintheLevystabilityregime.Theythenshowedthatinallsuchcases,theasymptoticformoftheconningpotentialV(x)isgivenasV(x)=lnx2+O(1=x)byusingtherelationbetweenthepotentialandthespectraldensityarisingintheCoulombgasanalogy.Aftertheidenticationoftheconningpotentialandthusthematrixensemble(freeLevymatrices),theyfurtherstudiedtheeigenvaluecorrelationsbasedonastandardRMTtechnique,namelytheorthogonalpolynomialmethod.ThismodelsuccessfullyincorporatestheasymptoticLevydistributionsandalsoallowsthecalculationsofeigenvaluecorrelationswithinthestandardRMTtechniquesduetotherotationalinvarianceoftheprobabilitymeasureoftheensemble.It,however,suggeststhatthe 3Togiveaquickillustrationofthemethod,considerthetrivialchoiceofR(z)=a,thenGreenfunctionG(z)=1 z)]TJ /F5 7.97 Tf 6.58 0 Td[(aandthusthespectraldensity(x)=(x)]TJ /F3 11.955 Tf 12.87 0 Td[(a).RefertoRef[ 22 ]formoredetails. 17 PAGE 18 twolevelcorrelationsoftheensembleshaveanontrivialNdependencethatcannotbesimplyscaledout. 1.3.3NonextensiveqEnsemblesOnewayofconstructingtheclassicalRMmodelsistomaximizetheShannonentropyS S)]TJ /F11 11.955 Tf 23.91 16.27 Td[(ZdHP(H)lnP(H),(1)subjecttotheconditionthattheprobabilityisnormalizedto1, ZdHP(H)=1.(1)HeredHisthemeasureassociatedwiththematrixelementsofH.Inparticular,requiringthatthevarianceofthematrixelementsbenite,(equivalentlyassumingthatextremelylargematrixelementsareimprobable) ZdHTr(H2)P(H)<1,(1)itcanbeshownbyusingtheLagrangemultiplier()methodthattheP(H)thatmaximizestheentropySsubjecttothetheaboveconditionsisgivenby P(H)dHe)]TJ /F8 7.97 Tf 6.59 0 Td[(TrH2dH.(1)Thus,themaximizationoftheentropywiththenitevarianceconstraintonP(H)leadstotheGaussianensembles.Inasimilarmanner,thenonextensiveqensembles[ 23 ]canbeconstructedfrommaximizingthenonextensiveentropySq Sq[P(H)]=1)]TJ /F11 11.955 Tf 11.96 9.64 Td[(RdH[P(H)]q q)]TJ /F6 11.955 Tf 11.96 0 Td[(1(1)withtheusualnormalizationcondition,RdHP(H)=1andqisaparameter.SimilartotheconstraintthatrequiresthenitevarianceofP(H)leadingtotheGaussian 18 PAGE 19 ensembles,thenonextensiveqensemblesrequirethefollowingcondition RdHtrH2[P(H)]q RdH[P(H)]q=2(1)withaconstant.UsingtheLagrangemultipliermethod,itcanbeshownthattheP(H)thatmaximizesthenonextensiveentropySqsubjecttotheaboveconstraintsisgivenby P(H)expq()]TJ /F4 11.955 Tf 9.29 0 Td[(trH2)(1)wheretheqexponentialfunctionexpq(x)isdenedas expq(x)f[1+(1)]TJ /F3 11.955 Tf 11.96 0 Td[(q)x]+g1 1)]TJ /F19 5.978 Tf 5.76 0 Td[(q(1)with [...]+=max(...,0).(1)Notethatforq=1,theexpq(x)reducestotheusualexponentialfunctionthusleadingtotheusualGaussianensembles.Thestudyshowsthatthenonextensiveqensemblesexhibitcharacteristicallydifferentbehaviordependingontheparameterq;forq>1,thedistributionsofeigenvaluedensityshowtruelongtailsandforq<1,thedistributionshavecompactsupport.However,thevariabilityoftheparameterqinthiscasedependsonthedimensionalityNoftheensemblesuchthatinthelargeNlimitwhereuniversalbehaviorisexpected,themaximumqallowedforthenonextensiveensemblesapproachesunity. 1.3.4EnsemblesBasedonSuperpositionsofClassicalorGaussianEnsemblesThereareseveralRMmodelscategorizedunderthisclassofensembles[ 24 28 ].Themainideaistoconstructgeneralizedensembleswiththesuperposition(ordeformation)oftheGaussianorWishartensembles.Toillustratethedetailsoftheidea,theapproachinRef.[ 24 ]willbediscussedinthefollowing.TheunderlyingideaoftheapproachintheRef.[ 24 ]istoworkwithanassociatedcharacteristicfunction,ratherthanaprobabilitydistribution,todiscoveranovelrandom 19 PAGE 20 matrixensemble.Itisshowninthisframeworkthatthegeneralnpointcorrelationfunctioncanbewrittendownas Rn(x1,,xn)=Z10dbf(b) (4b)n=2det[KGN(xi,xj)]i,j=1,2,...n(1)wheref(b)iscalledthespreadfunction,denedasanynonnegativenormalizablefunction,R10f(b)db=1andxi=xi=2p bandKGNreferstothewellknownsinekerneloftheGaussianensembles.Inparticular,thespectraldensity(1pointcorrelationfunction)canbegivenby N(x)=p 2N 2Z10dbf(b) b1=2KGN(x,x)(1)andtwopointclusterfunctiondenedasT2(x1,x2))]TJ /F3 11.955 Tf 22.16 0 Td[(R2(x1,x2)+R1(x1)R1(x2)hastheformT2=T02)]TJ /F4 11.955 Tf 11.95 0 Td[(T2,where T02(x1,x2)=Z10dbf(b) 4b[KGN(x1,x2)]2(1)and T=Z10dbf(b) 4bGN(x1)GN(x2))]TJ /F4 11.955 Tf 11.96 0 Td[(N(x1)N(x2)(1)Thus,avarietyofpossiblegeneralizedensemblescanbeobtainedfromaproperchoiceofsuperposition(achoiceofthespreadfunctionf(b))ofthekernelofGaussianensemble.Atthesametime,itispossibletostudythecorrelationfunctionsforthechoiceoff(b)immediately.Forexample,theGaussianensemblescanbethoughtofasatrivialexamplecorrespondingtothechoiceoff(b)=(b)]TJ /F3 11.955 Tf 12.49 0 Td[(b0).Foranontrivialchoiceoff(b)= \()b)]TJ /F7 7.97 Tf 6.59 0 Td[((+1)e)]TJ /F8 7.97 Tf 6.58 0 Td[(=b,itisshownthatthespectraldensityexhibitasymptoticpowerlawdistributions.AlthoughthisframeworkallowsasuccessfulrealizationofRM 20 PAGE 21 modelthatcanincorporatethedesiredfattailspectraldensity,itisnotyetcleariftheunfoldedtwolevelclusterfunctioncanhaveNindependentasymptoticlimit4. 4ItturnsoutthatthetwolevelcorrelationfunctioncarriesnontrivialNdependence.ThedetailswillbeprovidedinAPPENDIXB. 21 PAGE 22 CHAPTER2RANDOMMATRIXTHEORY 2.1RandomMatrixEnsemblesTherststeptoconstructaRMEistoconsiderasetofNNHermitianmatricesHwiththefollowingprobabilitymeasure PN(H)dH/e)]TJ /F5 7.97 Tf 6.59 0 Td[(TrV(H)dH,(2)whereV(H)isasuitablyincreasingfunctionsuchthattheprobabilitymeasureisnormalizable.TristhematrixtraceanddHtheinvariantmeasure.Especially,theTrV(H)ensurestherotationalinvarianceunderorthogonal,unitaryandsymplectictransformations.Moreexplicitly,itmeansthatunderatransformationR, H!M=RyHR,RyR=1(2)theprobabilitymeasureremainsthesame PN(H)dH=PN(M)dM.(2)Thus,byconstruction,allorientationsoftheeigenbasisareequallylikelyorinotherwords,thereisnopreferentialbasis.Eachofthesymmetriesdeterminesthestructureofthehamiltoniansystems,e.g., Orthogonalsymmetry(timereversalinvariantsystemswithrotationalsymmetry)!Realsymmetricmatrices Unitarysymmetry(systemsinwhichtimereversalsymmetryisbroken)!ComplexHermitianmatrices Symplecticsymmetry(timereversalinvariantsystemswithhalfintegerspinandbrokenrotationalsymmetry)!Selfdualquaternionmatrices.InordertondaspecicrealizationofaRME,oneneedstospecifytheprobabilitymeasureV(H)ofaRME.OnewaytondV(H)comesfromthemaximizationoftheentropyortheminimizationofinformationcontentoftheRME.Supposethatnothingis 22 PAGE 23 knownaboutthedetaileddynamicsofasystemotherthanthefundamentalsymmetries,whichgivesnoknowledgeaboutthematrixelementsofthehamiltoniansystemsotherthantheglobalsymmetries.Theinformation(I)containedintheNNHamiltonianmatrixHcanbedenedas IZdHP(H)lnP(H),(2)withtheconditionthattheprobabilityisnormalizedto1, ZdHP(H)=1.(2)requiringthevarianceofthematrixelementsbenite ZdHTr(H2)P(H)<1.(2)TheinformationinEq. 2 canbeminimizedsubjecttotheconstraintsinEq. 2 andEq. 2 byusingLagrangemultipliermethod.ForanarbitraryvariationP(H)ofP(H), I=ZdHP(H)f1+lnP(H)+Tr(H2)P(H)g=0(2)Thus, P(H)dH/e)]TJ /F8 7.97 Tf 6.59 0 Td[(Tr(H2)dH.(2)ThisistheprobabilitymeasureofthewellknownGaussianensembles.TheGaussianensemblescanalsobeobtainedbasedonthetwoassumptions;invarianceundertransformation(Orthogonal,Unitary,Symplectic)andstatisticalindependenceofmatrixelements[ 4 ].ThethreedifferentsymmetriesdenethreedifferentclassesofGaussianensembles:GaussianOrthogonalEnsemble(GOE),GaussianUnitaryEnsemble(GUE),andGaussianSymplecticEnsemble(GSE).IfweonlyrequiretherotationalinvarianceofP(H)dH,droppingtheassumptionofthestatisticalindependenceofthematrixelements,thenP(H)maybeanyfunctionofthetracesofpowersofH.Thus,ingeneral 23 PAGE 24 theprobabilitymeasurecanbewrittendownas P(H)dH/e)]TJ /F17 7.97 Tf 6.59 0 Td[(TrV(H)dH.(2)ItisalsopossibletoconsiderRMEswithnoninavarantformofprobabilitymeasureundertransformation,whichisthesocalledWignerclassensembles[ 35 ].Inthisclass,eachmatrixelementoftheensembleofNNmatrixarerandomlytakenfromi.i.d.probabilitydistributionssothatthegenericformoftheprobabilitymeasureofWignerensemblescanbegivenby P(M)dM/Yi,jf(Mi,j)Yi,jdMi,j(2)wheref(x)istheprobabilitydistributionofeachindividualmatrixelementandtheproductQi,jisperformedforallthedegreesofthefreedomoftheensembleunderconsideration.Someoftheexamplesare: WignerGuassianensembles:Hermitianmatriceswhoseentriesaregivenbyi.i.d.Gaussiandistributions. Levymatrices(WignerLevymatrices):Realsymmetricmatriceswhoseentriesaregivenbyi.i.d.Levydistributions. Bandedmatrices:Symmetricmatriceswhoseentriesmijarenonzeroforji)]TJ /F3 11.955 Tf 9.71 0 Td[(jj PAGE 25 NlimitiscentraltoestablishingtheuniversalityofRMEs.Thus,inthefollowing,wewillreviewtheorthogonalpolynomialmethodandtheresultsobtainedforclassicalandGaussianRMEsandthecriticalensembles. 2.2.1JointProbabilityDistributionFunction(JPDF)Inordertostudytheeigenvaluecorrelations,theprobabilitymeasureP(H)dHneedstobetransformedfromthematrixelementbasistotheeigenvalue/eigenvectorbasis, H!RyXRwithRyR=1.(2)Here,thematrixXisthediagonalmatrixcontainingeigenvalues(fxig,i=1,...,N)andthematrixRistherotationaltransformationcontainingeigenvectors.Afterthetransformationisconsidered,theinvariantmeasuredHinthematrixelementbasiscanberewrittenintheeigenvalue/eigenvectorbasis,e.g., dH=J(X,R)dXdR;i=1...N(2)whereJ(X,R)istheJacobianofthetransform.ThedXanddRrefertothemeasureassociatedwiththeeigenvaluesandparametersoftherotationaltransformationsrespectively.Here,theJacobianfactorcanbefurthercalculatedforagivensymmetryclassanditcanbeshownthat[ 4 ] J(X,R)=Yi PAGE 26 whereQi PAGE 27 Sincethedeterminantdoesnotchangebytheaddingtoanyoneroworcolumntomultiplesoftheotherrowsorcolumns,thematrixQcanberewrittenas Q=p0(x1)p w(x1)p0(x2)p w(x2)...p0(xN)p w(xN)p1(x1)p w(x1)p1(x2)p w(x2)...pN(xN)p w(xN)............pN)]TJ /F7 7.97 Tf 6.59 0 Td[(1(x1)p w(x1)pN)]TJ /F7 7.97 Tf 6.59 0 Td[(1(x2)p w(x2)...pN)]TJ /F7 7.97 Tf 6.58 0 Td[(1(xN)p w(xN)(2)wherepi(x)denotesanarbitrarypolynomialofdegreei.AndafterconsideringthematrixmultiplicationQTQ,wecanobtain PN(fxig)=1 N!detKN(x1,x1)KN(x1,x2)...KN(x1,xN)KN(x2,x1)KN(x2,x2)...KN(x2,xN)............KN(xN,x1)KN(xN,x2)...KN(xN,xN)(2)wherethetwolevelkernelKN(x,y)isdenedas KN(x,y)N)]TJ /F7 7.97 Tf 6.59 0 Td[(1Xi=0 i(x) i(y),(2)wherethewavefunctions i(x)pi(x)p w(x)=pi(x)e)]TJ /F5 7.97 Tf 6.59 0 Td[(V(x)=2.Herethechoicepi(x)isarbitrarybutifpi(x)ischosenasystemofthepolynomialsthatareorthogonalwithrespecttotheweightfunctionw(x)=e)]TJ /F5 7.97 Tf 6.59 0 Td[(V(x),i.e., 1Zpn(x)pm(x)w(x)dx=mn.(2)ThekernelKN(x,y)satisesfollowingimportantproperties. ZKN(x,x)dx=N(2)and KN(x,z)=ZKN(x,y)KN(y,z)dy.(2) 27 PAGE 28 Theseimplythat Zdet[KN(xi,yi)]i,j=1,...,mdxm=(N)]TJ /F3 11.955 Tf 11.95 0 Td[(m+1)det[KN(xi,xj)]i,j=1,...,m)]TJ /F7 7.97 Tf 6.59 0 Td[(1.(2) 2.2.3CorrelationFunctionsandClusterFunctionsTheprobabilityofndinganyneigenvaluesoutoftheavailableNeigenvaluesintheintervals,fxi!xi+dxi;i=1...ngisgivenbynlevelcorrelationfunction Rn(x1,...,xn)=N! (N)]TJ /F3 11.955 Tf 11.96 0 Td[(n)!Z1Z1dxn+1...dxNPN(x1,...,xN).(2)UsingthedeterminantformofJPDFandthepropertiesofthekernel,thenlevelcorrelationfunctioncanbewritteninacompactformgivenby Rn(x1,x2,....,xn)=det[KN(xi,xj)]fi,j=1,...,ng.(2)NotethatthediagonaltermsofthedeterminantaregivenbytheonelevelcorrelationfunctionR1(x)(ortheeigenvaluedensityN(x)) R1(x)=KN(x,x)=N)]TJ /F7 7.97 Tf 6.59 0 Td[(1Xi=0 i(x) i(y).(2)andtheoffdiagonaltermsofthedeterminantaregivenbytwolevelkernelintermsofwhichthetwolevelcorrelationfunctioncanbewrittendownas R2(x,y)=KN(x,x)KN(y,y))]TJ /F3 11.955 Tf 11.96 0 Td[(KN(x,y)KN(y,x).(2)Forpracticalpurpose,itisusefultointroducethenlevelclusterfunctionsdenedby Tn=XG()]TJ /F6 11.955 Tf 9.29 0 Td[(1)n)]TJ /F5 7.97 Tf 6.58 0 Td[(l(l)]TJ /F6 11.955 Tf 11.95 0 Td[(1)!lYi=1RGj(xj,withjinGk)(2)Here,Gstandsforanydivisionoftheindices(1,2,....n)intomsubgroups(G1,G2,...,Gm).Forexample,theonelevelclusterfunction T1(x)=R1(x)(2) 28 PAGE 29 andtwolevelclusterfunction T2(x,y)=)]TJ /F3 11.955 Tf 9.3 0 Td[(R2(x,y)+R1(x,x)R1(y,y) (2) =KN(x,y)2 (2) = N)]TJ /F7 7.97 Tf 6.58 0 Td[(1Xn=0 n(x) n(y)!2. (2) 2.2.4UnfoldingWithintheinvariantclassensembles,theonelevelcorrelationfunction(eigenvaluedensity)N(x)=KN(x,x)isnotuniversalanditisdependentonthespecicformoftheprobabilitymeasuredP(H).Sointhelimitoflargematrixsize,e.g.,N!1,weneedtointroduceaproperscalingvariabletostudytheuniversalityofcorrelationfunctionsuchthatinthescaledvariablethemeanspacingofeigenvaluesbecomeunity.Itisachievedbydeningtheunfoldingvariable dulimN!1KN(x,x)dx.(2)Theunfoldedtwolevelkernelcanbewrittenintermsoftheunfoldingvariables K(u,v)=limN!1KN(x,y) p KN(x,x)KN(y,y)(2)whereuandvaretheunfoldingvariables.Nowallthecorrelationfunctionsandtheclusterfunctionscanbewrittenintermsofthisunfoldedkernel.Forexample,unfoldedonelevelclusterfunctionsY(u)=1andthetwolevelclusterfunction Y2(u,v)=K(u,v)2(2)Thus,thestudyofcorrelationfunctionsrequiresunderstandingoftwolevelkernelintheproperlyscaledvariablewhichrequirestheknowledgeofasymptoticbehavioroftheorthogonalpolynomials. 29 PAGE 30 NotethatifthekernelsatisfyEq. 2 orinorderwords,thekernelisreproducing,theunfoldedtwolevelkernelsatisesthenormalizationofsumrule. ZdvY(u,v)=limN!1ZKN(y,y)KN(x,y)2 KN(x,x)KN(y,y)dy (2) =limN!11 KN(x,x)ZKN(x,y)KN(y,x)dy (2) =limN!1KN(x,x) KN(x,x)=1. (2) 2.3GaussianEnsembles 2.3.1SemiCirclelawFortheGaussianensemblesV(x)=x2,theeigenvaluedensityisgivenbythewellknownsemicirclelaw. GN(x)=KN(x,x)=Xi i(x) i(x)1 2p 2N)]TJ /F3 11.955 Tf 11.95 0 Td[(x2(2)wherethewavefunction n(x)=hn(x)e)]TJ /F5 7.97 Tf 6.59 0 Td[(x2=2satises Z1 n(x) m(y)e)]TJ /F5 7.97 Tf 6.59 0 Td[(x2dx=p 2nn!nm(2)wherehn(x)istheHermitepolynomialofdegreen.Thederivationofsemicirclelawcanbegiveninthefollowingway.Firstofall,werecognizethatEq. 2 ,thedeterminantformoftheJPDF,canbeconsideredastheprobabilitydensityofthegroundstateofthemanybodywavefunctionofnoninteractingfermions,e.g., P(x1,...,xN)/j(x1,...,xN)j2(2)where (x1,...,xN)= 0(x1) 0(x2)... 0(xN) 1(x1) 1(x1)... 1(x1)............ N)]TJ /F7 7.97 Tf 6.59 0 Td[(1(x1) N)]TJ /F7 7.97 Tf 6.59 0 Td[(1(x1)... N)]TJ /F7 7.97 Tf 6.58 0 Td[(1(x1)(2) 30 PAGE 31 Sincethewavefunctioninthedeterminantisthesolutionofonedimensionalharmonicoscillator,itsatisestheSchrodingerequationThus, )]TJ /F20 11.955 Tf 9.29 0 Td[(~d2 dx2+1 2x2 n(x)=(n+1=2)~ n(x).(2)Inthelargen=N)]TJ /F6 11.955 Tf 11.96 0 Td[(1limit,theFermimomentumcanbereadofflocallyatx pF=p 2N)]TJ /F6 11.955 Tf 11.96 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(x2.(2)ThedensityofonedimensionalfermionsisrelatedtotheFermimomentumpFby (x)1 2~ZpF)]TJ /F5 7.97 Tf 6.59 0 Td[(pFdp=pF ~=1.(2)CombiningEq. 2 andEq. 2 ,weget (x)1 p 2N)]TJ /F3 11.955 Tf 11.95 0 Td[(x2.(2) 2.3.2SineKernelInordertocalculatetheunfoldedtwolevelkernel,weconsiderthespectrumaroundtheorigininthelargeNlimit,socalleddoublescalinglimit(x!0andN!1limit)suchthatxp Nisnite.Theproperunfoldedvariableinthislimitdu=KN(x,x)dx=p 2N dx.Usingthedenitionofthekernel, KN(u,v)=limN!1 p 2NN)]TJ /F7 7.97 Tf 6.59 0 Td[(1Xi=0i( p 2Nu)i( p 2Nv).(2)RecallingtheChristoffelDarbouxformula, N)]TJ /F7 7.97 Tf 6.58 0 Td[(1Xj=0j(x)j(y)=r N 2hN(x)N)]TJ /F7 7.97 Tf 6.59 0 Td[(1(y))]TJ /F4 11.955 Tf 11.96 0 Td[(N(x)N)]TJ /F7 7.97 Tf 6.59 0 Td[(1(y) x)]TJ /F3 11.955 Tf 11.95 0 Td[(yi.(2) KN(u,v)=limN!1r N 2N(u p 2N)N)]TJ /F7 7.97 Tf 6.58 0 Td[(1(v p 2N))]TJ /F4 11.955 Tf 11.96 0 Td[(N(v p 2N)N)]TJ /F7 7.97 Tf 6.58 0 Td[(1(u p 2N) u)]TJ /F3 11.955 Tf 11.95 0 Td[(v(2) 31 PAGE 32 Usingasymptoticsolutionforharmonicoscillatorwavefunction, limm!1()]TJ /F6 11.955 Tf 9.3 0 Td[(1)mm1=42m(u p 2N)=)]TJ /F7 7.97 Tf 6.59 0 Td[(1=2cos(u)(2) limm!1()]TJ /F6 11.955 Tf 9.3 0 Td[(1)mm1=42m+1(u p 2N)=)]TJ /F7 7.97 Tf 6.58 0 Td[(1=2sin(u)(2)whereN=2m.Wegettheunfoldedkernel KG(u,v)=sin((u)]TJ /F3 11.955 Tf 11.95 0 Td[(v)) (u)]TJ /F3 11.955 Tf 11.95 0 Td[(v)(2)whichisthewellknownsinekernel. 2.3.3GapProbabilityandSpacingDistributionTheprobabilitythatthereisnoeigenvalueintheintervalofsiscalledgapprobabilitydenedasaFredholmdeterminant E(s)=det[1)]TJ /F6 11.955 Tf 13.73 2.65 Td[(KG](2)wheretheKGreferstothesinekernelovertheintervalof)]TJ /F3 11.955 Tf 9.3 0 Td[(s=2ands=2.Thespacingdistributionisrelatedtothegapprobabilityby p(s)=d2E(s) ds2(2)whichiscommonlyusedtostudyshortrangeuctuationsintheeigenvaluespectrum.Thisfunctionmeasurestheprobabilityofndingtwoneighboringeigenvaluesintheintervalofsintheunfoldedscale.Theanalyticalcalculationishighlynontrivialbutthegoodapproximationofthespacingdistributioncanbeobtainedfromconsidering22matrixmodelsuggestedbyWigner.Thegeneralexpressionforthespacingdistribution,Wigner'ssurmiseorWignerDysonstatistics, p(s)=cse)]TJ /F5 7.97 Tf 6.59 0 Td[(ds2(2)wherecanddareconstants.ThereferstothesymmetryparameteroftheGaussianensembles.Notethatinthesmallslimit,thespacingdistributionisdeterminedbys 32 PAGE 33 thatindicatesthatthetwoneighboringlevelsrepel.ThelevelrepulsionischaracteristicoftheGaussianensembles.ThisisincontrasttotheuncorrelatedorPossioncase,p(s)=e)]TJ /F5 7.97 Tf 6.59 0 Td[(s,inwhichthelevelrepulsionisabsentandthereforethelevelscanbebuncheduporseparatedfarapart. 2.3.4NumberVarianceThenumbervariance(s)=hn2i)262(hni2providesthemeasureofthelongrangeeigenvalueuctuation.Itisgivenby (s)=Zs=2)]TJ /F5 7.97 Tf 6.59 0 Td[(s=2Zs=2)]TJ /F5 7.97 Tf 6.59 0 Td[(s=2[R(u,v)]dudv (2) =Zs=2)]TJ /F5 7.97 Tf 6.59 0 Td[(s=2Zs=2)]TJ /F5 7.97 Tf 6.59 0 Td[(s=2[(u)]TJ /F3 11.955 Tf 11.95 0 Td[(v))]TJ /F3 11.955 Tf 11.95 0 Td[(Y(u,v)]dudv (2) whereR(u,v)istwolevelcorrelationfunctiondenedbyR(u,v)h(u)(v)i)]TJ /F6 11.955 Tf 18.79 0 Td[(1.UsingtheclusterfunctionofGUE,YG(u,v)=YG(u)]TJ /F3 11.955 Tf 11.95 0 Td[(v)=hsin((u)]TJ /F5 7.97 Tf 6.58 0 Td[(v)) (u)]TJ /F5 7.97 Tf 6.58 0 Td[(v)i2,thefurthercalculationgives G(s)=s)]TJ /F6 11.955 Tf 11.95 0 Td[(2Zs0(s)]TJ /F3 11.955 Tf 11.95 0 Td[(r)Y(r)dr (2) =1 2(ln(2s)++1)+O(s)]TJ /F7 7.97 Tf 6.59 0 Td[(1) (2) whereisEuler'sconstant,=0.5772....Notethatforlarges,G(s)/lns,whichdisplaysanothercharacteristicoftheGaussianEnsembles,thelevelrigidity.Incontrast,forthePoissonoruncorrelatedcase,RP(u,v)=(u)]TJ /F3 11.955 Tf 11.95 0 Td[(v)andthusP(s)=s. 2.4CriticalEnsembles 2.4.1AndersonTransitionandCriticalEnsemblesThesuggestionofanoveluniversalitybeyondtheGaussianensemblescomesfromthestudyoftheAndersontransitioninthedisorderedelectronicsystems[ 8 36 42 ].Inthesesystems,theGaussianensembleisonlyrelevantinthemetallicregimewherealltheeigenstatesareextendedacrosstheentiresystemandcorrelationsofthecorrespondingeigenvaluesarewelldescribedbytheWDstatistics.Asthedisorder 33 PAGE 34 ismadestrongenough,theeigenstatesbecomelocalizedandthustheeigenvaluesbecomeuncorrelated.Especiallyatthedelocalizationlocalizationtransition,ithasbeenestablishedthatthecorrelationsoftheeigenvectorsexhibitnovelfeatures[ 8 36 42 ]suchasmultifractalityandthecorrelationsoftheeigenvaluesleadtoalevelcompressibilitythatisintermediatebetweenWDandPoissonstatistics.Similarlyinthestudiesofquantumchaos,energylevelstatisticsofsystemsthatareintermediatebetweenchaoticandregularstatesalsorequiregeneralizationbeyondWDandPoissonstatistics[ 3 43 47 ].Inthesecontexts,extensivestudieshavebeencarriedouttoconstructaparametricgeneralizationofRMmodelsthatcrossoverfromWDtoPoisson[ 3 8 36 47 ]asafunctionoftheparameter.Someofthesegeneralizationsindeedcapturetheessentialfeaturesofthecriticalstatistics,amongwhichthefamilyofqRMEs[ 48 50 ]providesaparticularlyvaluableinsight.WithinthecommonframeworkofrotationallyinvariantRMmodels[ 4 ]theqRMEsshowhowtheuniversalityoftheGaussianensemblecharacterizedbythezeroparametertwolevelsinekernelbreaksdownandeventuallygivesrisetoadifferentkindofuniversalityforthecriticalensembles,characterizedbyaoneparametertwolevelsinhkernel.Inparticular,therotationallyinvariantRMmodelsarecharacterizedbyaconningpotentialwhichdenestheweightfunctionofasetoforthogonalpolynomials;thekeydifferencebetweentheGaussianandthecriticalensemblescomesfromthefactthatthecorrespondingorthogonalpolynomials,namelyclassicalvs.qorthogonalpolynomials,respectively,possessqualitativelydifferentasymptoticproperties[ 51 52 ]. 2.4.2CriticalStatisticsTheconjectureofthenoveluniversalityattheAndersontransitionisbasedonthestudiesofthespacingdistributionatthecriticalpointofthetightbindinghamiltonianon 34 PAGE 35 acubiclattice(LLL)givenby H=Xiicyici)]TJ /F11 11.955 Tf 11.96 11.36 Td[(Xi,jcyicj(2)wherecyiandciarethecreationandannihilationoperatorsatsitei,andjdenotestheneighborsitesofithusj=1,...,6onthecubiclattice;iistherandomenergyofthesiteiinunitofoverlapenergyofneighboringsitesthatisuniformlydistributedintherange[)]TJ /F3 11.955 Tf 9.3 0 Td[(W=2,W=2].Inparticular,Ref.[ 36 ]showedthatthespacingdistributionPC(s)isscaleinvariant(doesnotdependonthesystemsizeL)atthetransitionpointW=Wcbyinvestigatingthequantity(W,L)A)]TJ /F5 7.97 Tf 6.58 0 Td[(Aw Ap)]TJ /F5 7.97 Tf 6.58 0 Td[(Aw1;thesclaeinvariantPC(s)ischaracterizedbyalinearslopeinsfors1andanexponentialdecayfors1.ThusthenovelformofP(s)ishybridoftheWDstatisticsandthePoissionstatistics.AnothercharacteristicofthecriticalstatisticsisthenitelevelcompressibilityorsubPoissinannumbervariance,namelyC(s)=swith0<<1,whichisintermediatebetweentheWDstatisticsG(s)/lnsandPoissononeP(s)/s.Theoriginofthisbehaviorisknowntobethesumruleviolation.e.g., =dh(s)i ds=limL!1ZL=2)]TJ /F5 7.97 Tf 6.58 0 Td[(L=2R(s)ds=1)]TJ /F11 11.955 Tf 11.95 16.27 Td[(Z1Y(s)6=0.(2)whichisalsorelatedtothemultifractalnatureofthewavefunctioncorrelationatthecriticalpoint[ 8 36 42 ]. 2.5CoulombGasAnalogyDysonsuggestedthattheJPDFoftheinvariantensemblescanbewrittenasaformofGibbsdistributionforaclassicalonedimensionalsystemofNparticlesdescribedby 1whereAR12P(s)dsandlowerlimitoftheintegral2referstothecrossingpointofPp(s)andPw(s)occurringat2'2.002 35 PAGE 36 theHamiltonianH(fxig)[ 53 ]inthefollowingform. P(fxig)=1 ZN,exp[)]TJ /F4 11.955 Tf 9.3 0 Td[(H(fxig)],(2)where H(fxig)=)]TJ /F11 11.955 Tf 11.29 11.36 Td[(Xi PAGE 37 Thus,bytakingfunctionalderivative=V(y)inEq. 2 ,weget ZJdxh(x)ih(y)iR(x,z)lnjy)]TJ /F3 11.955 Tf 11.95 0 Td[(zj=)]TJ /F4 11.955 Tf 9.3 0 Td[()]TJ /F7 7.97 Tf 6.59 0 Td[(1(x)]TJ /F3 11.955 Tf 11.95 0 Td[(y).(2)InthelargeNlimit,aroundtheregionunderconsideration,ifh(x)iscalesasNandbecomesaconstant,wecanintroduceanewvariableu=x=scaledbythemeanlevelspacing=)]TJ /F7 7.97 Tf 6.59 0 Td[(1(i.e.,fortheGaussianensembles,(x)/p 2Ninthex!0limit)andrewriteEq. 2 ,whichreads ZJdwR(u)]TJ /F3 11.955 Tf 11.96 0 Td[(w)lnjv)]TJ /F3 11.955 Tf 11.95 0 Td[(wj=)]TJ /F4 11.955 Tf 9.3 0 Td[()]TJ /F7 7.97 Tf 6.59 0 Td[(1(u)]TJ /F3 11.955 Tf 11.95 0 Td[(v).(2)ThisimpliesthatthetwolevelcorrelationfunctiondoesnotdependonthespecicformofV(x),whichcarriesmicroscopicinformationofthesystem.Therefore,thetwolevelkernelisuniversal.Withinthisframework,itisunderstoodthatforthesoftconnementpotential,theaveragedensityisnotconstantintheN!1limitsothatitdoesnotsimplyscaleout.Therefore,thetwolevelcorrelationfunctionisexpectedtobedifferentfromtheWDuniversality[ 54 ]. 37 PAGE 38 CHAPTER3ENSEMBLESInthemotivationtoinvestigatetheuniversalityassociatedwithfattailorpowerlawRMEs,weintroducedafamilyofU(N)invariantrandommatrixensemblescharacterizedbyanasymptoticlogarithmicpotentialV(H)=A[lnH]1+with>0[ 55 56 ],namedensembles1.Thereasonforsuchsuggestionisbasedonthefollowingfewobservations.First,itisknownthatforV(H)/[lnH]2correspondingto=1limit(thecriticalensemble),theeigenvaluespectrumisgivenbyinversepowerlawdistribution,whichisknownbythemeaneldtheoreticapproach[ 57 ].Second,forV(H)'NlnHcorrespondingtothe!0limitwiththeconstantAbeingorderofN(freeLevymatrices),thespectraldensityisgivenbythefattaildistributions.Third,inthelimit1,itisexpectedthattheconnementpotentialmaygrowsufcientlystrong,therebyapproachingtheGaussianlimit.Therefore,wecanspeculatethattheparameterisacontrollingparameterofthepowerlawbehavior.Thefactthatsuchparametricgeneralization(generalizingthepowerofthelogarithmtoarbitraryrealvaluelargerthan1)connectstheexistingRMmodelsequippedwithrotationalinvarianceisinterestingsincethemodelallowsustoexploreanypossiblenoveluniversalityassociatedwithfattailRMEsaswellasthelogarithmicsoftconnementpotentialwithintheframeworkoftherotationallyinvariantRMT.ThegenericchoiceoftheconningpotentialV(x)thatgivesasymptoticlogarithmicbehaviorisV(x)=A[lnx]1+.However,ithasaunphysicalsingularityattheoriginsothatweneedtoregularizeitincertainway.Onepossiblewaytodoitischoosing,i.e.,V(x)=A[ln(1+x)]1+butthereareavarietyofotherformsthatdifferbytheregularizationbehaviorinthevicinityoforigin,whichwillnotchangethecharacteristics 1Theearliername`Levylikeensembles'inRef.[ 55 ]reectsthemotivationofthestudy. 38 PAGE 39 oftheensembles.Forourstudy,weparticularlychoosethefollowingformofthepotential, V(x)=1 ln(1=q)[sinh)]TJ /F7 7.97 Tf 6.59 0 Td[(1x]1+;>0;q<1.(3)Forsimplicity,wewillintroduceln(1=q)sothatthemodelhastwoparameterand.ThemeritofchoosingtheEq. 3 isthatinthelimit=1,itcoincideswiththeonepossibleformoftheweightfunctionoftheqRMEssothatwecancompareourresultswiththoseofqRMEs.FortheqRMEs,themathematicalpropertiesofthecorrespondingorthogonalpolynomials,knownastheIsmailMassonqpolynomials[ 52 ],arewellestablishedandaccordingly,thetwolevelkernel(sinhkernel)andtheallthespectralproperties(thecriticalstatistics)areverywellknown[ 48 49 ].Inthefollowingsections,wewillshowthatfor6=1,wecanconstructthecorrespondingorthogonalpolynomials,generalizationofqpolynomialsinarigorousnumericalmethodandthuswecanstudytheonelevel(spectraldensity)andthetwolevelkernelthatiscentraltothetestoftheuniversality.Wewillalsoshowthatthespectraldensityandthetwolevelcorrelationfunctionoftheensemblesexhibitnovelfeature;thespectraldensityisgivenbyapowerlawformandtwolevelkernelpossesstheanomalouscomponentwhichisconsideredoneofthecharacteristicsofthecriticalensembles.Inaddition,wewillsuggestanovelformoftwolevelkerneloftheensemblesbasedontheunfoldinganalysisanddiscussthedetailsofitsbehavioraswellasthepropertiesofthelevelstatisticsthatcanbededucedfromthekernel.Finally,wewilldiscusstheimplicationsandapplicationsoftheourresults. 3.1MethodThemaindifcultyofstudyingensemblesisthattheorthogonalpolynomialscorrespondingtotheweightfunctionw(x)=e)]TJ /F5 7.97 Tf 6.59 0 Td[(V(x)forthearbitraryvalues(expect=1)arenotknown.Thus,thersttaskistoobtaintheorthogonalpolynomialscorrespondingtotheweightfunctionoftheensembles.Inthefollowing,wewill 39 PAGE 40 reviewtheproceduretoconstructorthogonalpolynomialsfornontrivialarbitraryweightfunction.Tothisend,wedenetheorthogonalpolynomials(x)forarbitraryweightfunctionw(x),givenby Z1dxw(x)n(x)m(x)=n,mhn(3)wherehnisthenormalizationconstant.Itiswellknownthateveryorthogonalsystemofrealvaluedpolynomialssatisfyathreetermrecurrencerelation[ 51 ]2 xn(x)=n+1(x)+Snn(x)+Rnn)]TJ /F7 7.97 Tf 6.58 0 Td[(1(x),(3)whereSnandRnaretherealcoefcientsoftherecurrencerelation.Inparticular,Rnisrelatedtothenormalizationconstanthnbyhn+1=Rn+1hnForexample,theHermitepolynomialsaredeterminedbytherecurrencerelationof n+1(x)=xn(x))]TJ /F3 11.955 Tf 11.96 0 Td[(nn)]TJ /F7 7.97 Tf 6.59 0 Td[(1(x).(3)Therecurrencerelationoftheqpolynomialsisgivenby n+1(xjq)=xn(xjq))]TJ /F6 11.955 Tf 13.15 8.09 Td[(1 4q)]TJ /F5 7.97 Tf 6.58 0 Td[(n(1)]TJ /F3 11.955 Tf 11.96 0 Td[(qn)n)]TJ /F7 7.97 Tf 6.59 0 Td[(1(xjq),0 0.Wepointoutthatforthesecases,theweightfunctionisanevenfunction,namelyV()]TJ /F3 11.955 Tf 9.3 0 Td[(x)=V(x)andthus,allSn=0.Therefore,theRndeterminesthepropertiesoftheorthogonalpolynomials.Inparticular,bycomparingtheHermitepolynomialsandqpolynomials,weobservethat Rn/n(Hermitepolynomials) (3) /en(qpolynomials). (3) 2Here,weconsiderthatthen(x)isamonicpolynomial. 40 