arXiv subjects
Publications and source records attributed to Alexander Soshnikov.
Motivated by the Rudnick-Sarnak theorem we study limiting distribution of smoothed local correlations of the form $$ \sum_{j_1, j_2, \ldots, j_n} f(N\*(θ_{j_2}-θ_{j_1}), N\*(θ_{j_3}-θ_{j_1}), \ldots, N\*(θ_{j_n}-θ_{j_1}))$$ for the Circular United Ensemble of random matrices for sufficiently smooth test functions.
We study limiting distribution of pair counting statistics of the form $ \sum_{1\leq i\neq j\leq N} f(L_N\*(θ_i-θ_j))$ for the circular $β$-ensemble (C$β$E) of random matrices for sufficiently smooth test function $f$ and $L_N=O(N).$ For $β=2$ and $L_N=N$ our results are inspired by a classical result of Montgomery on pair correlation of zeros of Riemann zeta function.
We extend our results on the fluctuation of the pair counting statistic of the Circular Beta Ensemble $\sum_{i\neq j}f(L_N(θ_i-θ_j))$ for arbitrary $β>0$ in the mesoscopic regime $L_N=O(N^{2/3-ε})$.
We prove the Central Limit Theorem for finite-dimensional vectors of linear eigenvalue statistics of submatrices of Wigner random matrices under the assumption that test functions are sufficiently smooth. We connect the asymptotic covariance to a family of correlated Gaussian Free Fields.
We prove Gaussian fluctuation for pair counting statistics of the form $ \sum_{1\leq i\neq j\leq N} f(θ_i-θ_j)$ for the Circular Unitary Ensemble (CUE) of random matrices in the case of a slowly growing variance in the limit of large $N.$
In this paper, we consider a strongly-repelling model of $n$ ordered particles $\{e^{i θ_j}\}_{j=0}^{n-1}$ with the density $p({θ_0},\cdots, θ_{n-1})=\frac{1}{Z_n} \exp \left\{-\fracβ{2}\sum_{j \neq k} \sin^{-2} \left( \frac{θ_j-θ_k}{2}\right)\right\}$, $β>0$. Let $θ_j=\frac{2 πj}{n}+\frac{x_j}{n^2}+const$ such that $\sum_{j=0}^{n-1}x_j=0$. Define $ζ_n \left( \frac{2 πj}{n}\right) =\frac{x_j}{\sqrt{n}}$ and extend $ζ_n$ piecewise linearly to $[0, 2 π]$. We prove the functional convergence of $ζ_n(t)$ to $ζ(t)=\sqrt{\frac{2}β} \mathfrak{Re} \left( \sum_{k=1}^{\infty} \frac{1}{k} e^{ikt} Z_k \right)$, where $Z_k$ are i.i.d. complex standard Gaussian random variables.
We consider the limiting spectral distribution of matrices of the form $\frac{1}{2b_{n}+1} (R + X)(R + X)^{*}$, where $X$ is an $n\times n$ band matrix of bandwidth $b_{n}$ and $R$ is a non random band matrix of bandwidth $b_{n}$. We show that the Stieltjes transform of ESD of such matrices converges to the Stieltjes transform of a non-random measure. And the limiting Stieltjes transform satisfies an integral equation. For $R=0$, the integral equation yields the Stieltjes transform of the Marchenko-Pastur law.
In this paper, we study the fluctuation of linear eigenvalue statistics of Random Band Matrices defined by $M_{n}=\frac{1}{\sqrt{b_{n}}}W_{n}$, where $W_{n}$ is a $n\times n$ band Hermitian random matrix of bandwidth $b_{n}$, i.e., the diagonal elements and only first $b_{n}$ off diagonal elements are nonzero. Also variances of the matrix elmements are upto a order of constant. We study the linear eigenvalue statistics $\mathcal{N}(ϕ)=\sum_{i=1}^{n}ϕ(λ_{i})$ of such matrices, where $λ_{i}$ are the eigenvalues of $M_{n}$ and $ϕ$ is a sufficiently smooth function. We prove that $\sqrt{\frac{b_{n}}{n}}[\mathcal{N}(ϕ)-\mathbb{E} \mathcal{N}(ϕ)]\stackrel{d}{\to} N(0,V(ϕ))$ for $b_{n}>>\sqrt{n}$, where $V(ϕ)$ is given in the Theorem 1.
For fixed $m > 1$, we study the product of $m$ independent $N \times N$ elliptic random matrices as $N$ tends to infinity. Our main result shows that the empirical spectral distribution of the product converges, with probability $1$, to the $m$-th power of the circular law, regardless of the joint distribution of the mirror entries in each matrix. This leads to a new kind of universality phenomenon: the limit law for the product of independent random matrices is independent of the limit laws for the individual matrices themselves. Our result also generalizes earlier results of Götze-Tikhomirov and O'Rourke-Soshnikov concerning the product of independent iid random matrices.
We prove the Central Limit Theorem for linear statistics of the eigenvalues of band random matrices provided $\sqrt{n} \ll b_n \ll n$ and test functions are sufficiently smooth.
We show that beta ensembles in Random Matrix Theory with generic real analytic potential have the asymptotic equipartition property. In addition, we prove a Central Limit Theorem for the density of the eigenvalues of these ensembles.
Let $M_n$ be a $n \times n$ Wigner or sample covariance random matrix, and let $μ_1(M_n), μ_2(M_n), ..., μ_n(M_n)$ denote the unordered eigenvalues of $M_n$. We study the fluctuations of the partial linear eigenvalue statistics $$ \sum_{i=1}^{n-k} f(μ_i(M_n)) $$ as $n \rightarrow \infty$ for sufficiently nice test functions $f$. We consider both the case when $k$ is fixed and when $\min{k,n-k}$ tends to infinity with $n$.
We study the distribution of the outliers in the spectrum of finite rank deformations of Wigner random matrices. We assume that the matrix entries have finite fourth moment and extend the results by Capitaine, Donati-Martin, and Féral for perturbations whose eigenvectors are delocalized.
We study the fluctuations of the matrix entries of regular functions of Wigner random matrices in the limit when the matrix size goes to infinity. In the case of the Gaussian ensembles (GOE and GUE) this problem was considered by A.Lytova and L.Pastur in J. Stat. Phys., v.134, 147-159 (2009). Our results are valid provided the off-diagonal matrix entries have finite fourth moment, the diagonal matrix entries have finite second moment, and the test functions have four continuous derivatives in a neighborhood of the support of the Wigner semicircle law.
In this note, we extend the results about the fluctuations of the matrix entries of regular functions of Wigner random matrices obtained in arXiv:1103.3731 [math.PR] to Wigner matrices with non-i.i.d. entries provided certain Lindeberg type conditions for the fourth moments of the off-diagonal entries and the second moments of the diagonal entries are satisfied. In addition, we relax our conditions on the test functions and require that for some $s>3 \ \int (1+|k|)^{2s}\*|\hat{f}(k)|^2 \* dk <\infty.$
For fixed $m>1$, we consider $m$ independent $n \times n$ non-Hermitian random matrices $X_1, ..., X_m$ with i.i.d. centered entries with a finite $(2+η)$-th moment, $ η>0.$ As $n$ tends to infinity, we show that the empirical spectral distribution of $n^{-m/2} \*X_1 X_2 ... X_m$ converges, with probability 1, to a non-random, rotationally invariant distribution with compact support in the complex plane. The limiting distribution is the $m$-th power of the circular law.
We study the distribution of the outliers in the spectrum of finite rank deformations of Wigner random matrice under the assumption that the off-diagonal matrix entries have uniformly bounded fifth moment and the diagonal entries have uniformly bounded third moment. Using our recent results on the fluctuation of resolvent entries [31],[28], and ideas from [9], we extend results by M.Capitaine, C.Donati-Martin, and D.Féral [12], [13].
We extend the results about the fluctuations of the matrix entries of regular functions of Wigner matrices to the case of sample covariance random matrices.