Current Distribution and Random Matrix Ensembles for an Integrable Asymmetric Fragmentation Process

We investigate the statistical distribution of the zeros of Dirichlet L-functions both analytically and numerically. Using the Hardy-Littlewood conjecture about the distribution of primes we show that the two-point correlation function of these zeros coincides with that for eigenvalues of the Gaussian unitary ensemble of random matrices, and that the distributions of zeros of different L-functions are statistically independent. Applications of these results to Epstein's zeta functions are briefly discussed.

We generalize the recently discovered relationship between JT gravity and double-scaled random matrix theory to the case that the boundary theory may have time-reversal symmetry and may have fermions with or without supersymmetry. The matching between variants of JT gravity and matrix ensembles depends on the assumed symmetries. Time-reversal symmetry in the boundary theory means that unorientable spacetimes must be considered in the bulk. In such a case, the partition function of JT gravity is still related to the volume of the moduli space of conformal structures, but this volume has a quantum correction and has to be computed using Reidemeister-Ray-Singer "torsion." Presence of fermions in the boundary theory (and thus a symmetry $(-1)^F$) means that the bulk has a spin or pin structure. Supersymmetry in the boundary means that the bulk theory is associated to JT supergravity and is related to the volume of the moduli space of super Riemann surfaces rather than of ordinary Riemann surfaces. In all cases we match JT gravity or supergravity with an appropriate random matrix ensemble. All ten standard random matrix ensembles make an appearance -- the three Dyson ensembles and the seven Altland-Zirnbauer ensembles. To facilitate the analysis, we extend to the other ensembles techniques that are most familiar in the case of the original Wigner-Dyson ensemble of hermitian matrices. We also generalize Mirzakhani's recursion for the volumes of ordinary moduli space to the case of super Riemann surfaces.

There is considerable interest in understanding the relation between random-matrix ensembles and quantum chaotic systems in the context of the universality of energy-level correlations. In this connection, while Gaussian ensembles of random matrices have been studied extensively, not much is known about ensembles with non-Gaussian weight functions. Dyson has shown that the n-level correlation functions can be expressed in terms of a kernel function involving orthogonal and skew-orthogonal polynomials--orthogonal for matrix ensembles with unitary invariance and skew orthogonal for ensembles with orthogonal and symplectic invariances. We have obtained the following results. (1) Skew-orthogonal polynomials of both types are derived for the Jacobi class of weight functions including the limiting cases of associated Laguerre and Hermite (or Gaussian). (2) Matrix-integral representations are given for the general weight functions. (3) Asymptotic forms of the polynomials are obtained rigorously for the Jacobi class and in the form of an ansatz for the general case. (4) For the three types of ensembles, the (asymptotic) n-level correlation functions with appropriate scaling are shown to be universal, being independent of the weight function and location in the spectrum, and identical with the well-known Gaussian results. This provides a rigorous justification for the universality of the Gaussian ensemble results observed in quantum chaotic systems. As expected, the level density is not universal.

In the 70’s, Robert May demonstrated that complexity creates instability in generic models of ecological networks having random interaction matrices A. Similar random matrix models have since been applied in many disciplines. Central to assessing stability is the “circular law” since it describes the eigenvalue distribution for an important class of random matrices A. However, despite widespread adoption, the “circular law” does not apply for ecological systems in which density-dependence operates (i.e., where a species growth is determined by its density). Instead one needs to study the far more complicated eigenvalue distribution of the community matrix S = DA, where D is a diagonal matrix of population equilibrium values. Here we obtain this eigenvalue distribution. We show that if the random matrix A is locally stable, the community matrix S = DA will also be locally stable, providing the system is feasible (i.e., all species have positive equilibria D > 0). This helps explain why, unusually, nearly all feasible systems studied here are locally stable. Large complex systems may thus be even more fragile than May predicted, given the difficulty of assembling a feasible system. It was also found that the degree of stability, or resilience of a system, depended on the minimum equilibrium population.

In recent decades, determinants and Pfaffians were found for eigenvalue correlations of various random matrix ensembles. These structures simplify the average over a large number of ratios of characteristic polynomials to integrations over one and two characteristic polynomials only. Up to now it was thought that determinants occur for ensembles with Dyson index β = 2, whereas Pfaffians only for ensembles with β = 1, 4. We derive a non-trivial Pfaffian determinant for β = 2 random matrix ensembles which is similar to the one for β = 1, 4. Thus, it unveils a hidden universality of this structure. We also give a general relation between the orthogonal polynomials related to the determinantal structure and the skew-orthogonal polynomials corresponding to the Pfaffian. As a particular example, we consider the chiral unitary ensembles in great detail.

Random matrix theory and symmetric spaces

The unitary Wilson random matrix theory is an interpolation between the chiral Gaussian unitary ensemble and the Gaussian unitary ensemble. This new way of interpolation is also reflected in the orthogonal polynomials corresponding to such a random matrix ensemble. Although the chiral Gaussian unitary ensemble and the Gaussian unitary ensemble are associated with the Dyson index β = 2, the intermediate ensembles exhibit a mixing of orthogonal polynomials and skew-orthogonal polynomials. We consider the Hermitian and the non-Hermitian Wilson random matrix and derive the corresponding polynomials, their recursion relations, Christoffel–Darboux-like formulas, Rodrigues formulas and representations as random matrix averages in a unifying way. With the help of these results, we derive the unquenched k-point correlation function of the Hermitian and the non-Hermitian Wilson random matrix in terms of two-flavor partition functions only. This representation is due to a Pfaffian factorization. It drastically simplifies the expressions, which can be easily numerically evaluated. It also serves as a good starting point for studying the Wilson–Dirac operator in the ϵ-regime of lattice quantum chromodynamics.

We address the Primary User (PU) detection (spectrum sensing) problem, relevant to cognitive radio, from a finite random matrix theoretical (RMT) perspective. Specifically, we employ recently-derived closed-form and exact expressions for the distribution of the standard condition number (SCN) of uncorrelated and semi-correlated random dual central Wishart matrices of finite sizes in the design Hypothesis-Testing algorithms to detect the presence of PU signals. In particular, two algorithms are designed, with basis on the SCN distribution in the absence (H_0) and in the presence (H_1) of PU signals, respectively. Due to an inherent property of the SCN's, the H_0 test requires no estimation of SNR or any other information on the PU signal, while the H_1 test requires SNR only. Further attractive advantages of the new techniques are: a) due to the accuracy of the finite SCN distributions, superior performance is achieved under a finite number of samples, compared to asymptotic RMT-based alternatives; b) since expressions to model the SCN statistics both in the absence and presence of PU signal are used, the statistics of the spectrum sensing problem in question is completely characterized; and c) as a consequence of a) and b), accurate and simple analytical expressions for the receiver operating characteristic (ROC) — both in terms of the probability of detection as a function of the probability of false alarm (P_D versus P_F) and in terms of the probability of acquisition as a function of the probability of miss detection (P_A versus P_M) — are yielded. It is also shown that the proposed finite RMT-based algorithms outperform all similar alternatives currently known in the literature, at a substantially lower complexity. In the process, several new results on the distributions of eigenvalues and SCNs of random Wishart Matrices are offered, including a closed-form of the Marchenko-Pastur's Cumulative Density Function (CDF) and extensions of the latter, as well as variations of asymptotic the distributions of extreme eigenvalues (Tracy-Widom) and their ratio (Tracy-Widom-Curtiss), which are simpler than those obtained with the "spiked population model".

In this paper we apply the REFORM method[3, 4] to the deduction of the system of canonical equations for normalized spectral functions of the matrices A n + B n U n ( ε ) C n , where A n , B n and C n are nonrandom matrices, and U n ( ε ) is random matrix from class C 12 [26] or from the following class C 13 of distributions of random matrices: where are independent random real matrices whose entries are independent for every n, and for a certain δ > 0 the Lyapunov condition is fulfilled: This problem has been considered in some publications for matrices A n + U n , where Unitary matrix U n from the class C 1 [26] is distributed by Haar measure, at the “ad hoc” level, without a strong proof, on the basis of heuristic calculations. Therefore, the behavior of limit n.s.f. of the sum of a random Unitary matrix U n and a nonrandom matrix A n has not been discovered. Many conclusions of various kinds have been presented in the literature (see [13–27]) for random matrices A n + Ξ n B n , where Ξ n is the random matrix with independent entries, concerning, for example, the effectiveness of the REFORM method and the role of the martingale difference representation for the resolvent of random matrices. We give the formulation of the Circular Law for random matrices A n + B n U n ( ε ) C n , where A n , B n and C n are nonrandom matrices, and U n ( ε ) is from the so called Class C 12 of random Unitary matrices (see[26]). In this case the Circular law means that the support of accompanying spectral density p ( x, y ) of eigenvalues of A n + B n U n ( ε ) C n looks like several drops of mercury on a table, and inside of some drops it is possible that some dry circles appear. We call this support of limit density p ( x, y ) the Mercury support . If the distances between the centers of these drops are large enough we have several separate almost circular drops as for corresponding description of limit spectral density for the Hermitian random matrices [28, 29]. According to the distances between the centers of these drops, these drops might not touch each other or can merge creating fanciful shapes represented at the end of the paper, Figures 1–16. The analogy with the Circular Law is the following: for a simple Unitary matrix U n from class C 1 all its eigenvalues are distributed asymptotically uniformly on the circumference and, evidently, this limit distribution does not coincide with the uniform distribution on the circle. But if we will add to Unitary matrix U n any diagonal nonrandom matrix A n = ( δ ij a j ) then if the distances between diagonal entries a j are large enough we have almost the same picture of limit distribution of eigenvalues of U n + A n if Unitary matrix U n would be equal to the random matrix Ξ n with random independent entries satisfying the Global Circular Law (see Part I of this paper). The rough explanation of this phenomenon is the following: the expected scalar products of the vector row or vector column of matrix Ξ n have the order n −1 . Therefore, the matrix Ξ n looks like orthogonal random matrix. In this paper we use for matrices A n + B n U n ( ε ) C n the triply regularized V -transform[2, 5],( VICTORIA -transform of random matrix which is the abbreviation of the following words: Very Important Computational Transformation Of Randomly Independent Arrays ) where α > 0, ε ≠= 0, γ ≠= 0 are regularization parameters , τ = t + is is a complex number and µ n { x,G n ( t, s, γ, ε )} is the normalized spectral function of G -matrix: Such a V -transformation was used for the first time in 1982–1985 in [2,3,5]. Canonical equations for the limit distribution function of normalized spectral functions of some matrices A n + B n U n and the following V - domain G δ , δ > 0(see [27, pp.142, 173]) of the pseudodistribution of eigenvalues of matrices A n + B n U n are found: where are eigenvalues of the matrix.

We study the spectral properties of and spectral crossovers between different random matrix ensembles [Poissonian, Gaussian orthogonal ensemble (GOE), Gaussian unitary ensemble (GUE)] in correlated spin-chain systems, in the presence of random magnetic fields, and the scalar spin-chirality term, competing with the usual isotropic and time-reversal invariant Heisenberg term. We have investigated these crossovers in the context of the level-spacing distribution and the level-spacing ratio distribution. We use random matrix theory (RMT) analytical results to fit the observed Poissonian-to-GOE and GOE-to-GUE crossovers, and examine the relationship between the RMT crossover parameter $\ensuremath{\lambda}$ and scaled physical parameters of the spin-chain systems in terms of a scaling exponent. We find that the crossover behavior exhibits universality, in the sense that, it becomes independent of lattice size in the large Hamiltonian matrix dimension limit. Moreover, the scaling exponent obtained from such a finite size scaling analysis, seems to be quite robust and independent of the type of crossover considered or the specific spectral correlation measure used.

This chapter is devoted to statistical nuclear physics with SU(3) in the context of statistical nuclear spectroscopy and random matrix theory. First, some general aspects of trace propagation are described and trace propagation formulas for traces over SU(3) irreps, in SM and IBM spaces, in four different situations are given with applications to test the goodness of SU(3) symmetry in (2s1d) shell nuclei and also for understanding regularities generated by random interactions. Going beyond this, presented also are : (i) results for nuclear level density enhancement in deformed nuclei using the SU(3) symmetry of sdIBM-1; (ii) brief derivation of the asymptotic formula for the SU(3) multiplicities in the \(\{m\} \rightarrow (\lambda \mu )\) reduction for \(U((\eta +1)(\eta +2)/2) \supset SU(3)\); (iii) some results for the EGUE(2)-SU(3) random matrix ensemble with SU(3) symmetry.

We consider random normal matrix and planar symplectic ensembles, which can be interpreted as two-dimensional Coulomb gases having determinantal and Pfaffian structures, respectively. For a class of radially symmetric potentials with soft edges, we derive the asymptotic expansions of the log-partition functions up to and including the O(1)-terms as the number N of particles increases. Notably, our findings stress that the formulas of the $$O(\log N)$$ - and O(1)-terms in these expansions depend on the connectivity of the droplet. For random normal matrix ensembles, our formulas agree with the predictions proposed by Zabrodin and Wiegmann up to an additive constant depending on N but not on the background potential. For planar symplectic ensembles, the expansions contain a new kind of ingredient in the O(N)-terms, the logarithmic potential evaluated at the origin in addition to the entropy of the ensembles.

Let $\widetilde{X}_{M\times N}$ be a rectangular data matrix with independent real-valued entries $[\widetilde{x}_{ij}]$ satisfying $\mathbb {E}\widetilde{x}_{ij}=0$ and $\mathbb {E}\widetilde{x}^2_{ij}=\frac{1}{M}$, $N,M\to\infty$. These entries have a subexponential decay at the tails. We will be working in the regime $N/M=d_N,\lim_{N\to\infty}d_N\neq0,1,\infty$. In this paper we prove the edge universality of correlation matrices ${X}^{\dagger}X$, where the rectangular matrix $X$ (called the standardized matrix) is obtained by normalizing each column of the data matrix $\widetilde{X}$ by its Euclidean norm. Our main result states that asymptotically the $k$-point ($k\geq1$) correlation functions of the extreme eigenvalues (at both edges of the spectrum) of the correlation matrix ${X}^{\dagger}X$ converge to those of the Gaussian correlation matrix, that is, Tracy-Widom law, and, thus, in particular, the largest and the smallest eigenvalues of ${X}^{\dagger}X$ after appropriate centering and rescaling converge to the Tracy-Widom distribution. The asymptotic distribution of extreme eigenvalues of the Gaussian correlation matrix has been worked out only recently. As a corollary of the main result in this paper, we also obtain that the extreme eigenvalues of Gaussian correlation matrices are asymptotically distributed according to the Tracy-Widom law. The proof is based on the comparison of Green functions, but the key obstacle to be surmounted is the strong dependence of the entries of the correlation matrix. We achieve this via a novel argument which involves comparing the moments of product of the entries of the standardized data matrix to those of the raw data matrix. Our proof strategy may be extended for proving the edge universality of other random matrix ensembles with dependent entries and hence is of independent interest.

We develop a perturbative approach to study the supersymmetric nonlinear sigma model characterized by a generic coupling matrix in the strong coupling limit. The method allows us to calculate explicitly the moments of the eigenfunctions and the two-level correlation function in the lowest order of the perturbative expansion. We find that the obtained expressions are equivalent to the results derived before for the corresponding random matrix ensembles. Such an equivalence is elucidated and generalized to all orders of the perturbative expansion by mapping the sigma model onto the field theory describing the almost diagonal random matrices.

This paper focuses on large neural networks whose synaptic connectivity matrices are randomly chosen from certain random matrix ensembles. The dynamics of these networks can be characterized by the eigenvalue spectra of their connectivity matrices. In reality, neurons in a network do not necessarily behave in a similar way, but may belong to several different categories. The first study of the spectra of two-component neural networks was carried out by Rajan and Abbott [Phys. Rev. Lett. 97, 188104 (2006)]. In their model, neurons are either "excitatory" or "inhibitory," and strengths of synapses from different types of neurons have Gaussian distributions with different means and variances. A surprising finding by Rajan and Abbott is that the eigenvalue spectra of these types of random synaptic matrices do not depend on the mean values of their elements. In this paper we prove that this is true even for a much more general type of random neural network, where there is a finite number of types of neurons and their synaptic strengths have correlated distributions. Furthermore, using the diagrammatic techniques, we calculate the explicit formula for the spectra of synaptic matrices of multicomponent neural networks.