Related papers: The Limiting Eigenvalue Distribution of Iterated k…
For fixed positive integers m, we consider the product of m independent n by n random matrices with iid entries as in the limit as n tends to infinity. Under suitable assumptions on the entries of each matrix, it is known that the limiting…
Consider two types of products of independent random matrices, including products of Ginibre matrices and inverse Ginibre matrices and products of truncated Haar unitary matrices and inverse truncated Haar matrices. Each product matrix has…
Let A be a p-variate real Wishart matrix on n degrees of freedom with identity covariance. The distribution of the largest eigenvalue in A has important applications in multivariate statistics. Consider the asymptotics when p grows in…
The model of heavy Wigner matrices generalizes the classical ensemble of Wigner matrices: the sub-diagonal entries are independent, identically distributed along to and out of the diagonal, and the moments its entries are of order 1/N,…
We define the Euler number of a bipartite graph on $n$ vertices to be the number of labelings of the vertices with $1,2,...,n$ such that the vertices alternate in being local maxima and local minima. We reformulate the problem of computing…
We study the behavior of a real $p$-dimensional Wishart random matrix with $n$ degrees of freedom when $n,p\rightarrow\infty$ but $p/n\rightarrow 0$. We establish the existence of phase transitions when $p$ grows at the order…
We present a finer quantitative version of an observation due to Breuillard, Green, Guralnick and Tao which tells that for finite non-bipartite Cayley graphs, once the nontrivial eigenvalues of their normalized adjacency matrices are…
Random geometric graphs consist of randomly distributed nodes (points), with pairs of nodes within a given mutual distance linked. In the usual model the distribution of nodes is uniform on a square, and in the limit of infinitely many…
We consider the edge-triangle model, a two-parameter family of exponential random graphs in which dependence between edges is introduced through triangles. In the so-called replica symmetric regime, the limiting free energy exists together…
Suppose that $G_j$ is a sequence of finite connected planar graphs, and in each $G_j$ a special vertex, called the root, is chosen randomly-uniformly. We introduce the notion of a distributional limit $G$ of such graphs. Assume that the…
We consider a full rank deformation of the GUE $W_N+A_N$ where $A_N$ is a full rank Hermitian matrix of size $N$ and $W_N$ is a GUE. The empirical eigenvalue distribution $\mu_{A_N}$ of $A_N$ converges to a probability distribution $\nu$.…
Graph neural networks (GNNs) have been widely used in graph-related contexts. It is known that the separation power of GNNs is equivalent to that of the Weisfeiler-Lehman (WL) test; hence, GNNs are imperfect at identifying all…
We undertake an extensive numerical investigation of the graph spectra of thousands regular graphs, a set of random Erd\"os-R\'enyi graphs, the two most popular types of complex networks and an evolving genetic network by using novel…
A large i.i.d. random matrix with deterministic low-rank perturbation has been extensively studied, particularly in the aspects of the ESD (Empirical Spectral Distribution) and the outliers of eigenvalues. In this work, we investigate the…
We consider the empirical eigenvalue distribution of an $m\times m$ principle submatrix of an $n\times n$ random unitary matrix distributed according to Haar measure. Earlier work of Petz and R\'effy identified the limiting spectral measure…
In this paper, we derive the analytical behavior of the limiting spectral distribution of non-central covariance matrices of the "general information-plus-noise" type, as studied in [14]. Through the equation defining its Stieltjes…
The intention of the paper is to move a step towards a classification of network topologies that exhibit periodic quantum dynamics. We show that the evolution of a quantum system, whose hamiltonian is identical to the adjacency matrix of a…
We study the fluctuations of the eigenvalues of real valued large centrosymmetric random matrices via its linear eigenvalue statistic. This is essentially a central limit theorem (CLT) for sums of dependent random variables. The dependence…
Given a random quantum state of multiple distinguishable or indistinguishable particles, we provide an effective method, rooted in symplectic geometry, to compute the joint probability distribution of the eigenvalues of its one-body reduced…
Many solutions for scientific problems rely on finding the first (largest) eigenvalue and eigenvector of a particular matrix. We explore the distribution of the first eigenvector of a symmetric random sparse matrix. To analyze the…