Related papers: Eigenvalues of $p$-adic random matrices

We investigate the distribution of eigenvalues of weighted adjacency matrices from a specific ensemble of random graphs. We distribute $N$ vertices across a fixed number $\kappa$ of components, with asymptotically $\alpha_j \dot N$ vertices…

We study the eigenvalues of the covariance matrix $\frac{1}{n}M^*M$ of a large rectangular matrix $M=M_{n,p}=(\zeta_{ij})_{1\leq i\leq p;1\leq j\leq n}$ whose entries are i.i.d. random variables of mean zero, variance one, and having finite…

We develop further the graph limit theory for dense weighted graph sequences. In particular, we consider probability graphons, which have recently appeared in graph limit theory as continuum representations of weighted graphs, and we…

We study the random \'etale algebra generated by a random polynomial with i.i.d. coefficients distributed according to Haar measure normalized on $\mathbb{Z}_p$. We determine the probability that this random algebra is an unramified field,…

We define a class of "algebraic" random matrices. These are random matrices for which the Stieltjes transform of the limiting eigenvalue distribution function is algebraic, i.e., it satisfies a (bivariate) polynomial equation. The Wigner…

Consider the ensemble of real symmetric Toeplitz matrices, each independent entry an i.i.d. random variable chosen from a fixed probability distribution p of mean 0, variance 1, and finite higher moments. Previous investigations showed that…

This note starts from work done by Dai, Geary, and Kadanoff (Hui Dai, Zachary Geary, and Leo P. Kadanoff, H. Dai, Z. Geary and L. P. Kadanoff, Journal of Statistical Mechanics, P05012 (2009)) on exact eigenfunctions for Toeplitz operators.…

This article introduces a new kind of number systems on $p$-adic integers which is inspired by the well-known $3n+1$ conjecture of Lothar Collatz. A $p$-adic system is a piecewise function on $\mathbb{Z}_p$ which has branches for all…

This article focuses on the fluctuations of linear eigenvalue statistics of $T_{n\times p}T'_{n\times p}$, where $T_{n\times p}$ is an $n\times p$ Toeplitz matrix with real, complex or time-dependent entries. We show that as $n \rightarrow…

We suggest a method of analyzing the joint probability density (JPD) ${\cal P}_N(z,{\bf v})$ of an eigenvalue $z$ and the associated right eigenvector ${\bf v}$ (normalized with ${\bf v}^*{\bf v}=1$) for non-Hermitian random matrices of a…

We study Hermitian random matrix models with an external source matrix which has equispaced eigenvalues, and with an external field such that the limiting mean density of eigenvalues is supported on a single interval as the dimension tends…

We consider higher-dimensional generalizations of the normalized Laplacian and the adjacency matrix of graphs and study their eigenvalues for the Linial-Meshulam model $X^k(n,p)$ of random $k$-dimensional simplicial complexes on $n$…

Let $\mathcal A$ be the adjacency matrix of the Erd\H{o}s-R\'{e}nyi directed graph $\mathscr G(N,p)$. We denote the eigenvalues of $\mathcal A$ by $\lambda_1^{\cal A},...,\lambda^{\cal A}_N$, and $|\lambda_1^{\cal A}|=\max_i|\lambda_i^{\cal…

Consider the ensemble of Real Symmetric Toeplitz Matrices, each entry iidrv from a fixed probability distribution p of mean 0, variance 1, and finite higher moments. The limiting spectral measure (the density of normalized eigenvalues)…

For a prime number $p\ge 5$, we consider three classical cusp eigenforms $f_j(z)$ of weights $k_1, k_2, k_3$, of conductors $N_1, N_2, N_3$, and of nebentypus characters $\psi_j \bmod N_j$. According to H.Hida and R.Coleman, one can include…

We continue the study of operator algebras over the $p$-adic integers, initiated in our previous work [1]. In this sequel, we develop further structural results and provide new families of examples. We introduce the notion of $p$-adic von…

Given a collection $\{\lambda_1, \dots, \lambda_n\} $ of real numbers, there is a canonical probability distribution on the set of real symmetric or complex Hermitian matrices with eigenvalues $\lambda_1,\ldots,\lambda_n$. In this paper, we…

In this article, we establish a limiting distribution for eigenvalues of a class of auto-covariance matrices. The same distribution has been found in the literature for a regularized version of these auto-covariance matrices. The original…

Let $K$ be a $p$-adic local field. In this work we study a special kind of $p$-adic Galois representations of it. These representations are similar to the Galois representations occurred in the exceptional zero conjecture for modular forms.…

We propose a technique for calculating and understanding the eigenvalue distribution of sums of random matrices from the known distribution of the summands. The exact problem is formidably hard. One extreme approximation to the true density…