Related papers: Random matrix statistics and zeroes of $L$-functio…
We formulate an abstract notion of equidistribution for families of $\lambda$-probability spaces parameterized by admissible $\mathbb{Z}$-sets. Under the assumption of equidistribution, we show that the $\sigma$-moment generating functions…
Let $x$ be a complex random variable such that ${\E {x}=0}$, ${\E |x|^2=1}$, ${\E |x|^{4} < \infty}$. Let $x_{ij}$, $i,j \in \{1,2,...\}$ be independet copies of $x$. Let ${\Xb=(N^{-1/2}x_{ij})}$, $1\leq i,j \leq N$ be a random matrix.…
We study the distribution of singular values of product of random matrices pertinent to the analysis of deep neural networks. The matrices resemble the product of the sample covariance matrices, however, an important difference is that the…
Representation theory and the theory of symmetric functions have played a central role in Random Matrix Theory in the computation of quantities such as joint moments of traces and joint moments of characteristic polynomials of matrices…
There has recently been interest in relating properties of matrices drawn at random from the classical compact groups to statistical characteristics of number-theoretical L-functions. One example is the relationship conjectured to hold…
For $X$ a pre-$\lambda$ random variable, we show the $\sigma$-moment generating function of $-X$ can be obtained from the $\sigma$-moment generating function of $X$ by applying the composition of the standard and degree flip involutions on…
One of the main applications of free probability is to show that for appropriately chosen independent copies of $d$ random matrix models, any noncommutative polynomial in these $d$ variables has a spectral distribution that converges…
Random matrix theory has successfully modeled many systems in physics and mathematics, and often the analysis and results in one area guide development in the other. Hughes and Rudnick computed $1$-level density statistics for low-lying…
Random Matrix Theory (RMT) has successfully modeled diverse systems, from energy levels of heavy nuclei to zeros of $L$-functions. Many statistics in one can be interpreted in terms of quantities of the other; for example, zeros of…
The paper deals with distribution of singular values of product of random matrices arising in the analysis of deep neural networks. The matrices resemble the product analogs of the sample covariance matrices, however, an important…
In 1972 H. L. Montgomery announced a remarkable connection between the distribution of the zeros of the Riemann zeta-function and the distribution of eigenvalues of large random Hermitian matrices. Since then a number of startling…
We introduce a finite version of free probability for rectangular matrices that amounts to operations on singular values of polynomials. We show that we can replicate the transforms from free probability, and that asymptotically there is…
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…
We apply symmetric function theory to study random processes formed by singular values of products of truncations of Haar distributed symplectic and orthogonal matrices. These product matrix processes are degenerations of Macdonald…
We give a new heuristic for all of the main terms in the integral moments of various families of primitive L-functions. The results agree with previous conjectures for the leading order terms. Our conjectures also have an almost identical…
This is a survey article written for a workshop on L-functions and random matrix theory at the Newton Institute in July, 2004. The goal is to give some insight into how well-distributed sets of matrices in classical groups arise from…
We show that the Riemannian Gaussian distributions on symmetric spaces, introduced in recent years, are of standard random matrix type. We exploit this to compute analytically marginals of the probability density functions. This can be done…
First we survey generating function methods for obtaining useful probability estimates about random matrices in the finite classical groups. Then we describe a probabilistic picture of conjugacy classes which is coherent and beautiful.…
We consider products of independent large random rectangular matrices with independent entries. The limit distribution of the expected empirical distribution of singular values of such products is computed. The distribution function is…
Sigmoid functions play an important role in many areas of applied mathematics, including machine learning, population dynamics and probability. We place the study of sigmoid functions in the context of the derivative sub-group of the group…