Symmetric Random Matrices Research Articles

Central limit theorem for linear eigenvalue statistics of the adjacency matrices of random simplicial complexes

AbstractWe study the adjacency matrix of the Linial–Meshulam complex model, which is a higher‐dimensional generalization of the Erdős–Rényi graph model. Recently, Knowles and Rosenthal proved that the empirical spectral distribution of the adjacency matrix is asymptotically given by Wigner's semicircle law in a diluted regime. In this article, we prove a central limit theorem for the linear eigenvalue statistics for test functions of polynomial growth that is of class on a closed interval. The proof is based on higher‐dimensional combinatorial enumerations and concentration properties of random symmetric matrices. Furthermore, when the test function is a polynomial function, we obtain the explicit formula for the variance of the limiting Gaussian distribution.

Dynamical signatures of Chaos to integrability crossover in generalized random matrix ensembles

We introduce a two-parameter ensemble of generalized real symmetric random matrices called the β-Rosenzweig–Porter ensemble (β-RPE), parameterized by β, a fictitious inverse temperature of the analogous Coulomb gas model, and γ, controlling the relative strength of disorder. β-RPE encompasses RPE from all of the Dyson’s threefold symmetry classes: orthogonal, unitary and symplectic for . Firstly, we study the energy correlations by calculating the density and 2nd moment of the nearest neighbor spacing (NNS) and robustly quantify the crossover among various degrees of level repulsions. Secondly, the dynamical properties are determined from an exact calculation of the temporal evolution of the fidelity enabling an identification of the characteristic Thouless and the equilibration timescales. The relative depth of the correlation hole in the average fidelity serves as a dynamical signature of the crossover from chaos to integrability and enables us to construct the phase diagram of β-RPE in the γ-β plane. Our results are in qualitative agreement with numerically computed fidelity for matrix ensembles. Furthermore, we observe that for large N the 2nd moment of NNS and the relative depth of the correlation hole exhibit a second order phase transition at γ = 2.

On the distribution of equivalence classes of random symmetric p‐adic matrices

AbstractWe consider random symmetric matrices with independent entries distributed according to the Haar measure on for odd primes p and derive the distribution of their canonical form with respect to several equivalence relations. We give a few examples of applications including an alternative proof for the result of Bhargava, Cremona, Fisher, Jones and Keating on the probability that a random quadratic form over has a non‐trivial zero.

Singular value distribution of dense random matrices with block Markovian dependence

A block Markov chain is a Markov chain whose state space can be partitioned into a finite number of clusters such that the transition probabilities only depend on the clusters. Block Markov chains thus serve as a model for Markov chains with communities. This paper establishes limiting laws for the singular value distributions of the empirical transition matrix and empirical frequency matrix associated to a sample path of the block Markov chain whenever the length of the sample path is Θ(n2) with n the size of the state space.The proof approach is split into two parts. First, we introduce a class of symmetric random matrices with dependent entries called approximately uncorrelated random matrices with variance profile. We establish their limiting eigenvalue distributions by means of the moment method. Second, we develop a coupling argument to show that this general-purpose result applies to the singular value distributions associated with the block Markov chain.

Nonlinear Random Matrix-Based Intelligent Management Model for Swimming Place Waters

In this paper, a nonlinear random matrix approach is used to analyze the management of swimming place waters, and in this way, an intelligent management model is designed and applied to the actual swimming place waters management. Firstly, some basic small deviation results of the random matrix are presented. Then, several types of small deviation inequalities are obtained for the maximum eigenvalues of independent stochastic semi-positive definite (PSD) matrices. These small deviation inequalities are independent of the matrix dimension, and the results apply to high-dimensional and even infinite-dimensional matrices. For the inverse eigenvalue problem of higher-order doubly random matrices, the conclusion that two symmetric doubly random matrices are combined into a higher-order symmetric doubly random matrix is proved. In other words, the method of constructing new symmetric random matrices using smaller matrices with known spectra is applied to the inverse eigenvalue problem of higher-order symmetric random matrices, and sufficient conditions for the existence of the solution of the inverse eigenvalue problem of higher-order random matrices are derived, avoiding the discussion of the parity of the number of eigenvalues. This paper focuses on realistic management, to explore the standardization of safety management of swimming venues, and is based on modern management theory, risk management theory and methods, using research integration method, fieldwork method, questionnaire survey method, expert interview method, and other research methods, detailing the methods and conclusions of risk identification in safety management of swimming venues, and further exploring the safety management standards of swimming venues on this basis. And, combined with risk management theory, depending on the probability of occurrence and the degree of harm of risk, four basic disposal methods such as risk avoidance, risk control, risk transfer, and risk self-retaining are proposed.

Random symmetric matrices: rank distribution and irreducibility of the characteristic polynomial

AbstractConditional on the extended Riemann hypothesis, we show that with high probability, the characteristic polynomial of a random symmetric $\{\pm 1\}$ -matrix is irreducible. This addresses a question raised by Eberhard in recent work. The main innovation in our work is establishing sharp estimates regarding the rank distribution of symmetric random $\{\pm 1\}$ -matrices over $\mathbb{F}_p$ for primes $2 < p \leq \exp(O(n^{1/4}))$ . Previously, such estimates were available only for $p = o(n^{1/8})$ . At the heart of our proof is a way to combine multiple inverse Littlewood–Offord-type results to control the contribution to singularity-type events of vectors in $\mathbb{F}_p^{n}$ with anticoncentration at least $1/p + \Omega(1/p^2)$ . Previously, inverse Littlewood–Offord-type results only allowed control over vectors with anticoncentration at least $C/p$ for some large constant $C > 1$ .

Singularity of random symmetric matrices revisited

Let M n M_n be drawn uniformly from all ± 1 \pm 1 symmetric n × n n \times n matrices. We show that the probability that M n M_n is singular is at most exp ⁡ ( − c ( n log ⁡ n ) 1 / 2 ) \exp (-c(n\log n)^{1/2}) , which represents a natural barrier in recent approaches to this problem. In addition to improving on the best-known previous bound of Campos, Mattos, Morris and Morrison of exp ⁡ ( − c n 1 / 2 ) \exp (-c n^{1/2}) on the singularity probability, our method is different and considerably simpler: we prove a “rough” inverse Littlewood-Offord theorem by a simple combinatorial iteration.

Time dependent fluctuations of linear eigenvalue statistics of some patterned matrices

We consider the n × n reverse circulant and symmetric circulant random matrices with independent Brownian motion entries. With polynomial test functions ϕ, we discuss the joint fluctuation and tightness (in t and ϕ) of the time dependent linear eigenvalue statistics of these matrices as n → ∞ and show convergence to appropriate Gaussian processes. The proofs are mainly combinatorial.

Physics in nonfixed spatial dimensions via random networks.

We study the quantum statistical electronic properties of random networks which inherently lack a fixed spatial dimension. We use tools like the density of states (DOS) and the inverse participation ratio to uncover various phenomena, such as unconventional properties of the energy spectrum and persistent localized states (PLS) at various energies, corresponding to quantum phases with zero-dimensional (0D) and one-dimensional (1D) order. For small ratio of edges over vertices in the network R we find properties resembling graphene(honeycomb) lattices, like a similar DOS containing a linear dispersion relation at the band center at energy E=0. In addition, we find PLS at various energies including E=-1,0,1, and others, for example, related to the golden ratio. At E=0 the PLS lie at vertices that are not directly connected with an edge, due to partial bipartite symmetries of the random networks (0D order). At E=-1,1 the PLS lie mostly at pairs of vertices (bonds), while the rest of the PLS at other energies, like the ones related to the golden ratio, lie at lines of vertices of fixed length (1D order), at the spatial boundary of the network, resembling the edge states in confined graphene systems with zigzag edges. As the ratio R is increased the DOS of the network approaches the Wigner semicircle, corresponding to random symmetric matrices(Hamiltonians) and the PLS are reduced and gradually disappear as the connectivity in the network increases. Finally, we calculate the spatial dimension D of the network and its fluctuations. We obtain both integer and noninteger D and a logarithmic dependence on R. In addition, we examine the relation of D and its fluctuations to the electronic properties derived. Our results imply that universal physics can manifest in physical systems irrespectively of their spatial dimension. Relations to emergent spacetime in quantum and emergent gravity approaches are also discussed.

SUSY transfer matrix approach for the real symmetric 1d random band matrices

This paper adapts the recently developed rigorous application of the supersymmetric transfer matrix approach for the Hermitian 1d band matrices to the case of the orthogonal symmetry. We consider N×N block band matrices consisting of W×W random Gaussian blocks (parametrized by j,k∈Λ=[1,n]∩Z, N=nW) with a fixed entry’s variance Jjk=W−1(δj,k+βΔj,k) in each block. Considering the limit W,n→∞, we prove that the behaviour of the second correlation function of characteristic polynomials of such matrices in the bulk of the spectrum exhibit a crossover near the threshold W∼ N.

On the smallest singular value of symmetric random matrices

AbstractWe show that for an $n\times n$ random symmetric matrix $A_n$ , whose entries on and above the diagonal are independent copies of a sub-Gaussian random variable $\xi$ with mean 0 and variance 1, \begin{equation*}\mathbb{P}[s_n(A_n) \le \epsilon/\sqrt{n}] \le O_{\xi}(\epsilon^{1/8} + \exp(\!-\Omega_{\xi}(n^{1/2}))) \quad \text{for all } \epsilon \ge 0.\end{equation*} This improves a result of Vershynin, who obtained such a bound with $n^{1/2}$ replaced by $n^{c}$ for a small constant c, and $1/8$ replaced by $(1/8) - \eta$ (with implicit constants also depending on $\eta > 0$ ). Furthermore, when $\xi$ is a Rademacher random variable, we prove that \begin{equation*}\mathbb{P}[s_n(A_n) \le \epsilon/\sqrt{n}] \le O(\epsilon^{1/8} + \exp(\!-\Omega((\!\log{n})^{1/4}n^{1/2}))) \quad \text{for all } \epsilon \ge 0.\end{equation*} The special case $\epsilon = 0$ improves a recent result of Campos, Mattos, Morris, and Morrison, which showed that $\mathbb{P}[s_n(A_n) = 0] \le O(\exp(\!-\Omega(n^{1/2}))).$ Notably, in a departure from the previous two best bounds on the probability of singularity of symmetric matrices, which had relied on somewhat specialized and involved combinatorial techniques, our methods fall squarely within the broad geometric framework pioneered by Rudelson and Vershynin, and suggest the possibility of a principled geometric approach to the study of the singular spectrum of symmetric random matrices. The main innovations in our work are new notions of arithmetic structure – the Median Regularized Least Common Denominator (MRLCD) and the Median Threshold, which are natural refinements of the Regularized Least Common Denominator (RLCD)introduced by Vershynin, and should be more generally useful in contexts where one needs to combine anticoncentration information of different parts of a vector.

Cavity and replica methods for the spectral density of sparse symmetric random matrices

We review the problem of how to compute the spectral density of sparse symmetric random matrices, i.e. weighted adjacency matrices of undirected graphs. Starting from the Edwards-Jones formula, we illustrate the milestones of this line of research, including the pioneering work of Bray and Rodgers using replicas. We focus first on the cavity method, showing that it quickly provides the correct recursion equations both for single instances and at the ensemble level. We also describe an alternative replica solution that proves to be equivalent to the cavity method. Both the cavity and the replica derivations allow us to obtain the spectral density via the solution of an integral equation for an auxiliary probability density function. We show that this equation can be solved using a stochastic population dynamics algorithm, and we provide its implementation. In this formalism, the spectral density is naturally written in terms of a superposition of local contributions from nodes of given degree, whose role is thoroughly elucidated. This paper does not contain original material, but rather gives a pedagogical overview of the topic. It is indeed addressed to students and researchers who consider entering the field. Both the theoretical tools and the numerical algorithms are discussed in detail, highlighting conceptual subtleties and practical aspects.

Melonic dominance and the largest eigenvalue of a large random tensor

We consider a Gaussian rotationally invariant ensemble of random real totally symmetric tensors with independent normally distributed entries, and estimate the largest eigenvalue of a typical tensor in this ensemble by examining the rate of growth of a random initial vector under successive applications of a nonlinear map defined by the random tensor. In the limit of a large number of dimensions, we observe that a simple form of melonic dominance holds, and the quantity we study is effectively determined by a single Feynman diagram arising from the Gaussian average over the tensor components. This computation suggests that the largest tensor eigenvalue in our ensemble in the limit of a large number of dimensions is proportional to the square root of the number of dimensions, as it is for random real symmetric matrices.

Eigenvector statistics of Lévy matrices

We analyze statistics for eigenvector entries of heavy-tailed random symmetric matrices (also called Lévy matrices) whose associated eigenvalues are sufficiently small. We show that the limiting law of any such entry is non-Gaussian, given by the product of a normal distribution with another random variable that depends on the location of the corresponding eigenvalue. Although the latter random variable is typically nonexplicit, for the median eigenvector it is given by the inverse of a one-sided stable law. Moreover, we show that different entries of the same eigenvector are asymptotically independent, but that there are nontrivial correlations between eigenvectors with nearby eigenvalues. Our findings contrast sharply with the known eigenvector behavior for Wigner matrices and sparse random graphs.

Eigenvectors and controllability of non-Hermitian random matrices and directed graphs

We study the eigenvectors and eigenvalues of random matrices with iid entries. Let N be a random matrix with iid entries which have symmetric distribution. For each unit eigenvector v of N our main results provide a small ball probability bound for linear combinations of the coordinates of v. Our results generalize the works of Meehan and Nguyen [59] as well as Touri and the second author [67, 68, 69] for random symmetric matrices. Along the way, we provide an optimal estimate of the probability that an iid matrix has simple spectrum, improving a recent result of Ge [37]. Our techniques also allow us to establish analogous results for the adjacency matrix of a random directed graph, and as an application we establish controllability properties of network control systems on directed graphs.

Statistics of the spectral form factor in the self-dual kicked Ising model

We compute the full probability distribution of the spectral form factor in the self-dual kicked Ising model by providing an exact lower bound for each moment and verifying numerically that the latter is saturated. We show that at large enough times the probability distribution agrees exactly with the prediction of Random Matrix Theory if one identifies the appropriate ensemble of random matrices. We find that this ensemble is not the circular orthogonal one - composed of symmetric random unitary matrices and associated with time-reversal-invariant evolution operators - but is an ensemble of random matrices on a more restricted symmetric space (depending on the parity of the number of sites this space is either ${Sp(N)/U(N)}$ or ${O(2N)/{O(N)\!\times\!O(N)}}$). Even if the latter ensembles yield the same averaged spectral form factor as the circular orthogonal ensemble they show substantially enhanced fluctuations. This behaviour is due to a recently identified additional anti-unitary symmetry of the self-dual kicked Ising model.

Outliers in spectrum of sparse Wigner matrices

AbstractIn this paper, we study the effect of sparsity on the appearance of outliers in the semi‐circular law. Let be a sequence of random symmetric matrices such that each Wn is n × n with i.i.d. entries above and on the main diagonal equidistributed with the product , where is a real centered uniformly bounded random variable of unit variance and bn is an independent Bernoulli random variable with a probability of success pn. Assuming that , we show that for the random sequence given by , the ratio converges to one in probability. A noncentered counterpart of the theorem allows to obtain asymptotic expressions for eigenvalues of the Erdős–Renyi graphs, which were unknown in the regime . In particular, denoting by An the adjacency matrix of the Erdős–Renyi graph and by its kth largest (by the absolute value) eigenvalue, under the assumptions and we have (1) (No non‐trivial outliers): if then for any fixed k ≥ 2, converges to 1 in probability; and (2) (Outliers): if then there is ε > 0 such that for any , we have . On a conceptual level, our result reveals similarities in appearance of outliers in spectrum of sparse matrices and the so‐called BBP phase transition phenomenon in deformed Wigner matrices.

Random band matrices in the delocalized phase, III: averaging fluctuations

We consider a general class of symmetric or Hermitian random band matrices $$H=(h_{xy})_{x,y \in \llbracket 1,N\rrbracket ^d}$$ in any dimension $$d\ge 1$$ , where the entries are independent, centered random variables with variances $$s_{xy}=\mathbb {E}|h_{xy}|^2$$ . We assume that $$s_{xy}$$ vanishes if $$|x-y|$$ exceeds the band width W, and we are interested in the mesoscopic scale with $$1\ll W\ll N$$ . Define the generalized resolvent of H as $$G(H,Z):=(H - Z)^{-1}$$ , where Z is a deterministic diagonal matrix with entries $$Z_{xx}\in \mathbb {C}_+$$ for all x. Then we establish a precise high-probability bound on certain averages of polynomials of the resolvent entries. As an application of this fluctuation averaging result, we give a self-contained proof for the delocalization of random band matrices in dimensions $$d\ge 2$$ . More precisely, for any fixed $$d\ge 2$$ , we prove that the bulk eigenvectors of H are delocalized in certain averaged sense if $$N\le W^{1+\frac{d}{2}}$$ . This improves the corresponding results in He and Marcozzi (Diffusion profile for random band matrices: a short proof, 2018. arXiv:1804.09446 ) that imposed the assumption $$N\ll W^{1+\frac{d}{d+1}}$$ , and the results in Erdős and Knowles (Ann Henri Poincare12(7):1227–1319, 2011; Commun Math Phys 303(2): 509–554, 2011) that imposed the assumption $$N\ll W^{1+\frac{d}{6}}$$ . For 1D random band matrices, our fluctuation averaging result was used in Bourgade et al. (J Stat Phys 174:1189–1221, 2019; Random band matrices in the delocalized phase, I: quantum unique ergodicity and universality, 2018. arXiv:1807.01559 ) to prove the delocalization conjecture and bulk universality for random band matrices with $$N\ll W^{4/3}$$ .

Spectral rigidity of non-Hermitian symmetric random matrices near the Anderson transition

We study the spectral rigidity of the non-Hermitian analog of the Anderson model suggested by Tzortzakakis, Makris and Economou (TME). This is a $L\times L \times L$ tightly bound cubic lattice, where both real and imaginary parts of on-site energies are independent random variables uniformly distributed between $-W/2$ and $W/2$. The TME model may be used to describe a random laser. In a recent paper we proved that this model has the Anderson transition at $W= W_c \simeq 6$ in three dimension. Here we numerically diagonalize TME $L \times L \times L$ cubic lattice matrices and calculate the number variance of eigenvalues in a disk of their complex plane. We show that on the metallic side $W < 6$ of the Anderson transition, complex eigenvalues repel each other as strongly as in the complex Ginibre ensemble only in a disk containing $N_c(L,W)$ eigenvalues. We find that $N_c(L,W)$ is proportional to $L$ and grows with decreasing $W$ similarly to the number of energy levels $N_c$ in the Thouless energy band of the Anderson model.

Random Band Matrices in the Delocalized Phase I: Quantum Unique Ergodicity and Universality

Consider $N\times N$ symmetric one-dimensional random band matrices with general distribution of the entries and band width $W \geq N^{3/4+\varepsilon}$ for any $\varepsilon>0$. In the bulk of the spectrum and in the large $N$ limit, we obtain the following results. (i) The semicircle law holds up to the scale $N^{-1+\varepsilon}$ for any $\varepsilon>0$. (ii) The eigenvalues locally converge to the point process given by the Gaussian orthogonal ensemble at any fixed energy. (iii) All eigenvectors are delocalized, meaning their ${\rm L}^\infty$ norms are all simultaneously bounded by $N^{-\frac{1}{2}+\varepsilon}$ (after normalization in ${\rm L}^2$) with overwhelming probability, for any $\varepsilon>0$. (iv )Quantum unique ergodicity holds, in the sense that the local ${\rm L}^2$ mass of eigenvectors becomes equidistributed with overwhelming probability. We extend the mean-field reduction method \cite{BouErdYauYin2017}, which required $W=\Omega(N)$, to the current setting $W \ge N^{3/4+\varepsilon}$. Two new ideas are: (1) A new estimate on the generalized resolvent of band matrices when $W \geq N^{3/4+\varepsilon}$. Its proof, along with an improved fluctuation average estimate, will be presented in parts 2 and 3 of this series \cite {BouYanYauYin2018,YanYin2018}. (2) A strong (high probability) version of the quantum unique ergodicity property of random matrices. For its proof, we construct perfect matching observables of eigenvector overlaps and show they satisfying the eigenvector moment flow equation \cite{BouYau2017} under the matrix Brownian motions.