Let $A$ be an $n\times n$ random matrix with independent rows $R_1(A),\dots,R_n(A)$, and assume that for any $i\leq n$ and any three-dimensional linear subspace $F\subset {\mathbb R}^n$ the orthogonal projection of $R_i(A)$ onto $F$ has distribution density $ρ(x):F\to{\mathbb R}_+$ satisfying $ρ(x)\leq C_1/\max(1,\|x\|_2^{2000})$ ($x\in F$) for some constant $C_1>0$. We show that for any fixed $n\times n$ real matrix $M$ we have $${\mathbb P}\{s_{\min}(A+M)\leq t n^{-1/2}\}\leq C'\, t,\quad\quad t>0,$$ where $C'>0$ is a universal constant. In particular, the above result holds if the rows of $A$ are independent centered log-concave random vectors with identity covariance matrices. Our method is free from any use of covering arguments, and is principally different from a standard approach involving a decomposition of the unit sphere and coverings, as well as an approach of Sankar-Spielman-Teng for non-centered Gaussian matrices.
The problem of estimating the smallest singular value of random square matrices is important in connection with matrix computations and analysis of the spectral distribution. In this survey, we consider recent developments in the study of quantitative invertibility in the non-Hermitian setting, and review some applications of this line of research.
Let $σ_n(\cdot)$ denote the least singular value of a $n \times n$ matrix. It is well-known that $\mathbb{P}[σ_n(A) \le \varepsilon] \le \varepsilon n$ if $A$ is drawn from the real Ginibre ensemble of $n \times n$ matrices and $\mathbb{P}[σ_n(A) \le \varepsilon] \le \varepsilon^2 n^2$ if $A$ is drawn from the complex Ginibre ensemble. In this paper, we will show a similar phenomenon occurs for sparse random matrices.
We show that a perturbation of any fixed square matrix D by a random unitary matrix is well invertible with high probability. A similar result holds for perturbations by random orthogonal matrices; the only notable exception is when D is close to orthogonal. As an application, these results completely eliminate a hard-to-check condition from the Single Ring Theorem by Guionnet, Krishnapur and Zeitouni.
Let $A=(a_{ij})$ be an $n\times n$ random matrix with i.i.d. entries such that $\mathbb{E} a_{11} = 0$ and $\mathbb{E} {a_{11}}^2 = 1$. We prove that for any $δ>0$ there is $L>0$ depending only on $δ$, and a subset $\mathcal{N}$ of $B_2^n$ of cardinality at most $\exp(δn)$ such that with probability very close to one we have $$A(B_2^n)\subset \bigcup_{y\in A(\mathcal{N})}\bigl(y+L\sqrt{n}B_2^n\bigr).$$ As an application, we show that for some $L'>0$ and $u\in[0,1)$ depending only on the distribution law of $a_{11}$, the smallest singular value $s_n$ of the matrix $A$ satisfies $\mathbb{P}\{s_n…
We consider a class of sparse random matrices of the form $A_n =(ξ_{i,j}δ_{i,j})_{i,j=1}^n$, where $\{ξ_{i,j}\}$ are i.i.d.~centered random variables, and $\{δ_{i,j}\}$ are i.i.d.~Bernoulli random variables taking value $1$ with probability $p_n$, and prove a quantitative estimate on the smallest singular value for $p_n = Ω(\frac{\log n}{n})$, under a suitable assumption on the spectral norm of the matrices. This establishes the invertibility of a large class of sparse matrices. For $p_n =Ω( n^{-α})$ with some $α\in (0,1)$, we deduce that the condition number of $A_n$ is of order $n$ with prob…
The Restricted Invertibility problem is the problem of selecting the largest subset of columns of a given matrix $X$, while keeping the smallest singular value of the extracted submatrix above a certain threshold. In this paper, we address this problem in the simpler case where $X$ is a random matrix but with the additional constraint that the selected columns be almost orthogonal to a given vector $v$. Our main result is a lower bound on the number of columns we can extract from a normalized i.i.d. Gaussian matrix for the worst $v$.
We study n by n symmetric random matrices H, possibly discrete, with iid above-diagonal entries. We show that H is singular with probability at most exp(-n^c), and the spectral norm of the inverse of H is O(sqrt{n}). Furthermore, the spectrum of H is delocalized on the optimal scale o(n^{-1/2}). These results improve upon a polynomial singularity bound due to Costello, Tao and Vu, and they generalize, up to constant factors, results of Tao and Vu, and Erdos, Schlein and Yau.
Let $d\geq 3$ be a fixed integer and $A$ be the adjacency matrix of a random $d$-regular directed or undirected graph on $n$ vertices. We show there exist constants $\mathfrak d>0$, \begin{align*} {\mathbb P}(\text{$A$ is singular in $\mathbb R$})\leq n^{-\mathfrak{d}}, \end{align*} for $n$ sufficiently large. This answers an open problem by Frieze [12] and Vu [28,19]. The key idea is to study the singularity probability of adjacency matrices over a finite field ${\mathbb F}_p$. The proof combines a local central limit theorem and a large deviation estimate.
We prove two basic conjectures on the distribution of the smallest singular value of random n times n matrices with independent entries. Under minimal moment assumptions, we show that the smallest singular value is of order n^{-1/2}, which is optimal for Gaussian matrices. Moreover, we give a optimal estimate on the tail probability. This comes as a consequence of a new and essentially sharp estimate in the Littlewood-Offord problem: for i.i.d. random variables X_k and real numbers a_k, determine the probability P that the sum of a_k X_k lies near some number v. For arbitrary coefficients a_k …
In this paper, we investigate the invertibility of sparse symmetric matrices. We show that for an $n\times n$ sparse symmetric random matrix $A$ with $A_{ij} = δ_{ij} ξ_{ij}$ is invertible with high probability. Here, $δ_{ij}$s, $i\ge j$ are i.i.d. Bernoulli random variables with $\mathbb{P} \left(ξ_{ij}=1 \right) =p \ge n^{-c}$, $ξ_{ij}, i\ge j$ are i.i.d. random variables with mean 0, variance 1 and finite forth moment $M_4$, and $c$ is constant depending on $M_4$. More precisely, $$ s_{\rm min} (A) > \varepsilon \sqrt{\frac{p}{n}}. $$ with high probability.
Let $d\geq 3$ be a fixed integer, and a prime number $p$ such that $\gcd(p,d)=1$. Let $A$ be the adjacency matrix of a random $d$-regular directed graph on $n$ vertices. We show that as a random matrix in ${\mathbb F}_p$, \begin{equation} {\mathbb P}(\text{$A$ is singular in ${\mathbb F}_p$})\leq \frac{1+{\mathrm{o}}(1)}{p-1}, \end{equation} as $n$ goes to infinity. As a consequence, as a random matrix in $\mathbb R$, \begin{equation} {\mathbb P}(\text{$A$ is singular in $\mathbb R$})={\mathrm{o}}(1) \end{equation} as $n$ goes to infinity. This answers an open problem by Frieze [12] and Vu [29…
We consider three different models of sparse random graphs:~undirected and directed Erdős-Rényi graphs, and random bipartite graph with an equal number of left and right vertices. For such graphs we show that if the edge connectivity probability $p \in (0,1)$ satisfies $n p \ge \log n + k(n)$ with $k(n) \to \infty$ as $n \to \infty$, then the adjacency matrix is invertible with probability approaching one (here $n$ is the number of vertices in the two former cases and the number of left and right vertices in the latter case). If $np \le \log n -k(n)$ then these matrices are invertible with pro…
Let $p \in (0,1/2)$ be fixed, and let $B_n(p)$ be an $n\times n$ random matrix with i.i.d. Bernoulli random variables with mean $p$. We show that for all $t \ge 0$, \[\mathbb{P}[s_n(B_n(p)) \le tn^{-1/2}] \le C_p t + 2n(1-p)^{n} + C_p (1-p-ε_p)^{n},\] where $s_n(B_n(p))$ denotes the least singular value of $B_n(p)$ and $C_p, ε_p > 0$ are constants depending only on $p$. In particular, \[\mathbb{P}[B_{n}(p) \text{ is singular}] = 2n(1-p)^{n} + C_{p}(1-p-ε_p)^{n},\] which confirms a conjecture of Litvak and Tikhomirov. We also confirm a conjecture of Nguyen by showing that if $Q_{n}$ is an $n\ti…
We study invertibility of matrices of the form $D+R$ where $D$ is an arbitrary symmetric deterministic matrix, and $R$ is a symmetric random matrix whose independent entries have continuous distributions with bounded densities. We show that $|(D+R)^{-1}| = O(n^2)$ with high probability. The bound is completely independent of $D$. No moment assumptions are placed on $R$; in particular the entries of $R$ can be arbitrarily heavy-tailed.
We study the lower tail behavior of the least singular value of an $n\times n$ random matrix $M_n := M+N_n$, where $M$ is a fixed complex matrix with operator norm at most $\exp(n^{c})$ and $N_n$ is a random matrix, each of whose entries is an independent copy of a complex random variable with mean $0$ and variance $1$. Motivated by applications, our focus is on obtaining bounds which hold with extremely high probability, rather than on the least singular value of a typical such matrix. This setting has previously been considered in a series of influential works by Tao and Vu, most notably in …
In this article, we consider random Wigner matrices, that is symmetric matrices such that the subdiagonal entries of Xn are independent, centered, and with variance one except on the diagonal where the entries have variance two. We prove that, under some suitable hypotheses on the laws of the entries, the law of the largest eigenvalue satisfies a large deviation principle with the same rate function as in the Gaussian case. The crucial assumption is that the Laplace transform of the entries must be bounded above by the Laplace transform of a centered Gaussian variable with same variance. This …
Consider a doubly-infinite array of iid centered variables with moment conditions, from which one can extract a finite number of rectangular, overlapping submatrices, and form the corresponding Wishart matrices. We show that under basic smoothness assumptions, centered linear eigenstatistics of such matrices converge jointly to a Gaussian vector with an interesting covariance structure. This structure, which is similar to those appearing in work of Borodin, Borodin and Gorin, and Johnson and Pal can be described in terms of the height function, and leads to a connection with the Gaussian Free …
The probability of the small deviations of the matrix $AA^T$ determinant is estimated, where $A$ is an $n\times\infty$ random matrix with centered entries having joint Gaussian distribution. The inequality obtained is sharp in a sence.
We obtain lower tail estimates for the smallest singular value of random matrices with independent but non-identically distributed entries. Specifically, we consider $n\times n$ matrices with complex entries of the form \[ M = A\circ X + B = (a_{ij}ξ_{ij} + b_{ij}) \] where $X=(ξ_{ij})$ has iid centered entries of unit variance and $A$ and $B$ are fixed matrices. In our main result we obtain polynomial bounds on the smallest singular value of $M$ for the case that $A$ has bounded (possibly zero) entries, and $B= Z\sqrt{n}$ where $Z$ is a diagonal matrix with entries bounded away from zero. As …
In this work, we study the properties of a pentadiagonal symmetric matrix with perturbed corners. More specifically, we present explicit expressions for characterizing when this matrix is non-negative and positive definite in two special and important cases. We also give a closed expression for the determinant of such matrices. Previous works present the determinant in a recurrence form but not in an explicit one. As an application of these results, we also study the limiting cumulant generating function associated to the bivariate sequence of random vectors (n^{-1} (\sum_{k=1}^n X_k^2 , \sum_…
In this article, we study the fluctuations of the random variable: $$ {\mathcal I}_n(ρ) = \frac 1N \log\det(Σ_n Σ_n^* + ρI_N),\quad (ρ>0) $$ where $Σ_n= n^{-1/2} D_n^{1/2} X_n\tilde D_n^{1/2} +A_n$, as the dimensions of the matrices go to infinity at the same pace. Matrices $X_n$ and $A_n$ are respectively random and deterministic $N\times n$ matrices; matrices $D_n$ and $\tilde D_n$ are deterministic and diagonal, with respective dimensions $N\times N$ and $n\times n$; matrix $X_n=(X_{ij})$ has centered, independent and identically distributed entries with unit variance, either real or comple…
Let $X,X_1,...,X_n$ be independent identically distributed random variables. In this paper we study the behavior of the concentration functions of the weighted sums $\sum\limits_{k=1}^{n}a_k X_k$ with respect to the arithmetic structure of coefficients $a_k$. Such concentration results recently became important in connection with investigations about singular values of random matrices. In this paper we formulate and prove some refinements of a result of Vershynin (R. Vershynin, Invertibility of symmetric random matrices, arXiv:1102.0300. (2011). Published in Random Structures and Algorithms, v…
We give a non-asymptotic bound on the spectral norm of a $d\times d$ matrix $X$ with centered jointly Gaussian entries in terms of the covariance matrix of the entries. In some cases, this estimate is sharp and removes the $\sqrt{\log d}$ factor in the noncommutative Khintchine inequality.
In this paper, we prove that an $n\times n$ matrix $A$ with independent centered subgaussian entries satisfies \[ s_{n+1-l}(A) \le C_1t \frac{l}{\sqrt{n}} \] with probability at least $1-\exp(-C_2tl)$. This yields $s_{n-l}(A) \sim \frac{cl}{\sqrt{n}}$ in combination with a known lower bound. These results can be generalized to the rectangular matrix case.
We prove that the logarithm of the determinant of a Wigner matrix satisfies a central limit theorem in the limit of large dimension. Previous results about fluctuations of such determinants required that the first four moments of the matrix entries match those of a Gaussian. Our work treats symmetric and Hermitian matrices with centered entries having the same variance and subgaussian tail. In particular, it applies to symmetric Bernoulli matrices and answers an open problem raised by Tao and Vu.
Consider a matrix $Σ_n$ with random independent entries, each non-centered with a separable variance profile. In this article, we study the limiting behavior of the random bilinear form $u_n^* Q_n(z) v_n$, where $u_n$ and $v_n$ are deterministic vectors, and Q_n(z) is the resolvent associated to $Σ_n Σ_n^*$ as the dimensions of matrix $Σ_n$ go to infinity at the same pace. Such quantities arise in the study of functionals of $Σ_n Σ_n^*$ which do not only depend on the eigenvalues of $Σ_n Σ_n^*$, and are pivotal in the study of problems related to non-centered Gram matrices such as central limi…
In this paper we obtain a Bernstein type inequality for the sum of self-adjoint centered and geometrically absolutely regular random matrices with bounded largest eigenvalue. This inequality can be viewed as an extension to the matrix setting of the Bernstein-type inequality obtained by Merlevède et al. (2009) in the context of real-valued bounded random variables that are geometrically absolutely regular. The proofs rely on decoupling the Laplace transform of a sum on a Cantor-like set of random matrices.
We consider a square random matrix of size $N$ of the form $P(Y,A)$ where $P$ is a noncommutative polynomial, $A$ is a tuple of deterministic matrices converging in $\ast$-distribution, when $N$ goes to infinity, towards a tuple $a$ in some $\mathcal{C}^*$-probability space and $Y$ is a tuple of independent matrices with i.i.d. centered entries with variance $1/N$. We investigate the eigenvalues of $P(Y,A)$ outside the spectrum of $P(c,a)$ where $c$ is a circular system which is free from $a$. We provide a sufficient condition to guarantee that these eigenvalues coincide asymptotically with th…
We consider products of independent square random non-Hermitian matrices. More precisely, let $n\geq 2$ and let $X_1,\ldots,X_n$ be independent $N\times N$ random matrices with independent centered entries with variance $N^{-1}$. It was shown by Götze and Tikhomirov and by Soshnikov and O'Rourke that the limit of the empirical spectral distribution of the product $X_1\cdots X_n$ is supported in the unit disk. We prove that if the entries of the matrices $X_1,\ldots,X_n$ satisfy uniform subexponential decay condition, then the spectral radius of $X_1\cdots X_n$ converges to 1 almost surely as $…
Consider a $N\times n$ random matrix $Y_n=(Y_{ij}^{n})$ where the entries are given by $$ Y_{ij}^{n}=\frac{σ_{ij}(n)}{\sqrt{n}} X_{ij}^{n} $$ the $X_{ij}^{n}$ being centered, independent and identically distributed random variables with unit variance and $(σ_{ij}(n); 1\le i\le N, 1\le j\le n)$ being an array of numbers we shall refer to as a variance profile. We study in this article the fluctuations of the random variable $$ \log\det(Y_n Y_n^* + ρI_N) $$ where $Y^*$ is the Hermitian adjoint of $Y$ and $ρ> 0$ is an additional parameter. We prove that when centered and properly rescaled, this r…
We prove that the (non-symmetric) adjacency matrix of a uniform random $d$-regular directed graph on $n$ vertices is asymptotically almost surely invertible, assuming $\min(d,n-d)\ge C\log^2n$ for a sufficiently large constant $C>0$. The proof makes use of a coupling of random regular digraphs formed by "shuffling" the neighborhood of a pair of vertices, as well as concentration results for the distribution of edges recently obtained by the author (arXiv:1410.5595). We also apply our general approach to prove a.a.s.\ invertibility of Hadamard products $Σ\circ Ξ$, where $Ξ$ is a matrix of iid u…
For large dimensional non-Hermitian random matrices $X$ with real or complex independent, identically distributed, centered entries, we consider the fluctuations of $f(X)$ as a matrix where $f$ is an analytic function around the spectrum of $X$. We prove that for a generic bounded square matrix $A$, the quantity $\mathrm{Tr}f(X)A$ exhibits Gaussian fluctuations as the matrix size grows to infinity, which consists of two independent modes corresponding to the tracial and traceless parts of $A$. We find a new formula for the variance of the traceless part that involves the Frobenius norm of $A$ …
We consider a Wigner-type ensemble, i.e. large hermitian $N\times N$ random matrices $H=H^*$ with centered independent entries and with a general matrix of variances $S_{xy}=\mathbb E|H_{xy}|^2$. The norm of $H$ is asymptotically given by the maximum of the support of the self-consistent density of states. We establish a bound on this maximum in terms of norms of powers of $S$ that substantially improves the earlier bound $2\| S\|^{1/2}_\infty$ given in [arXiv:1506.05098]. The key element of the proof is an effective Markov chain approximation for the contributions of the weighted Dyck paths a…
We study largest singular values of large random matrices, each with mean of a fixed rank $K$. Our main result is a limit theorem as the number of rows and columns approach infinity, while their ratio approaches a positive constant. It provides a decomposition of the largest $K$ singular values into the deterministic rate of growth, random centered fluctuations given as explicit linear combinations of the entries of the matrix, and a term negligible in probability. We use this representation to establish asymptotic normality of the largest singular values for random matrices with means that ha…
For a class of symmetric random matrices whose entries are martingale differences adapted to an increasing filtration, we prove that under a Lindeberg-like condition, the empirical spectral distribution behaves asymptotically similarly to a corresponding matrix with independent centered Gaussian entries having the same variances. Under a slightly reinforced condition, the approximation holds in the almost sure sense. We also point out several sufficient regularity conditions imposed to the variance structure for convergence to the semicircle law or the Marchenko-Pastur law and other convergenc…
We show that various identities from [1] and [3] involving Gould-Hopper polynomials can be deduced from the real but also complex orthogonal invariance of multivariate Gaussian distributions. We also deduce from this principle a useful stochastic representation for the inner product of two non-centered Gaussian vectors and two non-centered Gaussian matrices. [1] J. Daboul, S. S. Mizrahi, O(N) symmetries, sum rules for generalized Hermite polynomials and squeezed state, J. Phys. A: Math. Gen. 38 (2005) 427-448 [3] P. Graczyk, A. Nowak, A composition formula for squares of Hermite polynomials an…
We consider Green's functions $G(z):=(H-z)^{-1}$ of Hermitian random band matrices $H$ on the $d$-dimensional lattice $(\mathbb Z/L\mathbb Z)^d$. The entries $h_{xy}=\bar h_{yx}$ of $H$ are independent centered complex Gaussian random variables with variances $s_{xy}=\mathbb E|h_{xy}|^2$. The variances satisfy a banded profile so that $s_{xy}$ is negligible if $|x-y|$ exceeds the band width $W$. For any $n\in \mathbb N$, we construct an expansion of the $T$-variable, $T_{xy}=|m|^2 \sum_αs_{xα}|G_{αy}|^2$, with an error $O(W^{-nd/2})$, and use it to prove a local law on the Green's function. Th…
Let $X= \{X(t), t \in \mathbb R^N\}$ be a centered Gaussian random field with values in $\mathbb R^d$ satisfying certain conditions and let $F \subset \mathbb R^d$ be a Borel set. In our main theorem, we provide a sufficient condition for $F$ to be polar for $X$, i.e. $\mathbb P \big( X(t) \in F \hbox{ for some } t \in \mathbb R^N \big) = 0$, which improves significantly the main result in Dalang et al [7], where the case of $F$ being a singleton was considered. We provide a variety of examples of Gaussian random field for which our result is applicable. Moreover, by using our main theorem, we…
We consider the moment space $\mathcal{M}^{p}_{2n+1}$ of moments up to the order $2n + 1$ of $p_n\times p_n$ real matrix measures defined on the interval $[0,1]$. The asymptotic properties of the Hankel determinant $\{\log\det (M_{i+j}^{p_n})_{i,j=0,\ldots,\lfloor nt\rfloor}\}_{t\in [0,1]}$ of a uniformly distributed vector $(M_1,\dots ,M_{2n+1})^t\sim\mathcal{U}(\mathcal{M}_{2n+1})$ are studied when the dimension $n$ of the moment space and the size of the matrices $p_n$ converge to infinity. In particular weak convergence of an appropriately centered and standardized version of this process …
We consider the local eigenvalue distribution of large self-adjoint $N\times N$ random matrices $\mathbf{H}=\mathbf{H}^*$ with centered independent entries. In contrast to previous works the matrix of variances $s_{ij} = \mathbb{E}\, |h_{ij}|^2 $ is not assumed to be stochastic. Hence the density of states is not the Wigner semicircle law. Its possible shapes are described in the companion paper [1]. We show that as $N$ grows, the resolvent, $\mathbf{G}(z)=(\mathbf{H}-z)^{-1}$, converges to a diagonal matrix, $ \mathrm{diag}(\mathbf{m}(z)) $, where $\mathbf{m}(z)=(m_1(z),\dots,m_N(z))$ solves …
Consider a sample of a centered random vector with unit covariance matrix. We show that under certain regularity assumptions, and up to a natural scaling, the smallest and the largest eigenvalues of the empirical covariance matrix converge, when the dimension and the sample size both tend to infinity, to the left and right edges of the Marchenko--Pastur distribution. The assumptions are related to tails of norms of orthogonal projections. They cover isotropic log-concave random vectors as well as random vectors with i.i.d. coordinates with almost optimal moment conditions. The method is a refi…
We study the spectral norm of N-dimensional hermitian random matrices whose entries are zero outside of the band of the width b along the principal diagonal. Inside this band the elements are given by gaussian centered jointly independent random variables with the variance of the order 1/b. We show that if b tends to infinity faster than the third power of log N, then the spectral norm is bounded with probability 1.
For large random matrices $X$ with independent, centered entries but not necessarily identical variances, the eigenvalue density of $XX^*$ is well-approximated by a deterministic measure on $\mathbb{R}$. We show that the density of this measure has only square and cubic-root singularities away from zero. We also extend the bulk local law in [arXiv:1606.07353] to the vicinity of these singularities.
We show that the spectral radius of an $N\times N$ random symmetric matrix with i.i.d. bounded centered but non-symmetrically distributed entries is bounded from above by $ 2 \*σ+ o(N^{-6/11+ε}), $ where $σ^2 $ is the variance of the matrix entries and $ε$ is an arbitrary small positive number. Our bound improves the earlier results by Z.Füredi and J.Komlós (1981), and the recent bound obtained by Van Vu (2005).
We show that the spectral radius of an $N\times N$ random symmetric matrix with i.i.d. bounded centered but non-symmetrically distributed entries is bounded from below by $ 2 \*σ- o(N^{-6/11+ε}), $ where $σ^2 $ is the variance of the matrix entries and $ε$ is an arbitrary small positive number. Combining with our previous result from [7], this proves that for any $ε>0, $ one has $$ \|A_N\| =2 \*σ+ o(N^{-6/11+ε}) $$ with probability going to 1 as $N \to \infty. $
We study a new random matrix ensemble $X$ which is constructed by an application of a two dimensional linear filter to a matrix of iid random variables with infinite fourth moments. Our result gives asymptotic lower and upper bounds for the spectral norm of the (centered) sample covariance matrix $XX^\T$ when the number of columns as well es the number of rows of $X$ tend to infinity.
We consider products of independent random matrices taken from the induced Ginibre ensemble with complex or quaternion elements. The joint densities for the complex eigenvalues of the product matrix can be written down exactly for a product of any fixed number of matrices and any finite matrix size. We show that the squared absolute values of the eigenvalues form a permanental process, generalising the results of Kostlan and Rider for single matrices to products of complex and quaternionic matrices. Based on these findings, we can first write down exact results and asymptotic expansions for th…
Let $M_n$ be a class of symmetric sparse random matrices, with independent entries $M_{ij} = δ_{ij} ξ_{ij}$ for $i \leq j$. $δ_{ij}$ are i.i.d. Bernoulli random variables taking the value $1$ with probability $p \geq n^{-1+δ}$ for any constant $δ> 0$ and $ξ_{ij}$ are i.i.d. centered, subgaussian random variables. We show that with high probability this class of random matrices has simple spectrum (i.e. the eigenvalues appear with multiplicity one). We can slightly modify our proof to show that the adjacency matrix of a sparse Erdős-Rényi graph has simple spectrum for $n^{-1+δ} \leq p \leq 1- n…
