Related papers: In support of $n$-correlation

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…

We derive the double scaling limit of eigenvalue correlations in the random matrix model at critical points and we relate the limiting correlation functions to a nonlinear hierarchy of ordinary differential equations.

Spectral properties of random matrices play an important role in statistics, machine learning, communications, and many other areas. Engaging results regarding the convergence of the empirical spectral distribution (ESD) and the…

We study the spectral properties of a rank-one multiplicative perturbation of a unitary matrix, a model introduced by Fyodorov. Building upon earlier results by Forrester and Ipsen, we provide a direct proof that the eigenvalues converge to…

We give an expository account of the theory of intertwining operators for connected reductive $p$--adic groups, and their connection with automorphic $L$--functions. Our purpose is to illustrate the relation between harmonic analysis and…

The closure of a discrete exponential family is described by a finite set of equations corresponding to the circuits of an underlying oriented matroid. These equations are similar to the equations used in algebraic statistics, although they…

We observe that the Laplacian of a random graph G on N vertices represents and explicitly solvable model in the limit of infinitely increasing N. Namely, we derive recurrent relations for the limiting averaged moments of the adjacency…

We give a randomness-efficient homomorphism test in the low soundness regime for functions, $f: G\to \mathbb{U}_t$, from an arbitrary finite group $G$ to $t\times t$ unitary matrices. We show that if such a function passes a derandomized…

Negative association for a family of random variables $(X_i)$ means that for any coordinate--wise increasing functions $f,g$ we have $$\E f(X_{i_1},...,X_{i_k}) g(X_{j_1},...,X_{j_l}) \leq \E f(X_{i_1},...,X_{i_k}) \E…

We consider eigenvalue condition numbers and backward errors for a class of symmetric nonlinear eigenvalue problems with eigenvector nonlinearities. For both of these quantities, we derive explicit and computable expressions that can be…

We compute the auto-correlations functions of order $m\ge 1$ for the characteristic polynomials of random matrices from certain subgroups of the unitary groups $\U(2)$ and $\U(3)$ by applying branching rules. These subgroups can be…

This paper establishes the universality of parametric correlations of eigenfunctions in chaotic and weakly disordered systems. We demonstrate this universality in the framework of the gaussian random matrix process and obtain predictions…

We review our recent results on pseudo-hermitian random matrix theory which were hitherto presented in various conferences and talks. (Detailed accounts of our work will appear soon in separate publications.) Following an introduction of…

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…

The eigendecomposition of a matrix is the central procedure in probabilistic models based on matrix factorization, for instance principal component analysis and topic models. Quantifying the uncertainty of such a decomposition based on a…

A theorem on computation of the homological Conley index of an isolated invariant set of the Poincar\'e map associated to a section in a rotating local dynamical system $\phi$ is proved. Let $(N,L)$ be an index pair for a discretization…

Consider a finite sequence of independent random permutations, chosen uniformly either among all permutations or among all matchings on n points. We show that, in probability, as n goes to infinity, these permutations viewed as operators on…

We present a unified description of extremal metrics for the Laplace and Steklov eigenvalues on manifolds of arbitrary dimension using the notion of $n$-harmonic maps. Our approach extends the well-known results linking extremal metrics for…

It has been observed that the statistical distribution of the eigenvalues of random matrices possesses universal properties, independent of the probability law of the stochastic matrix. In this article we find the correlation functions of…

We study the problem of approximating the eigenspectrum of a symmetric matrix $\mathbf A \in \mathbb{R}^{n \times n}$ with bounded entries (i.e., $\|\mathbf A\|_{\infty} \leq 1$). We present a simple sublinear time algorithm that…