Related papers: The Discrepancy of Random Rectangular Matrices
We consider random stochastic matrices $M$ with elements given by $M_{ij}=|U_{ij}|^2$, with $U$ being uniformly distributed on one of the classical compact Lie groups or associated symmetric spaces. We observe numerically that, for large…
We give a probabilistic introduction to determinantal and permanental point processes. Determinantal processes arise in physics (fermions, eigenvalues of random matrices) and in combinatorics (nonintersecting paths, random spanning trees).…
I present here some results on the statistical behaviour of large random matrices in an ensemble where the probability distribution is not a function of the eigenvalues only. The perturbative expansion can be cast in a closed form and the…
We introduce deterministic perturbation schemes for the recently proposed random directions stochastic approximation (RDSA) [17], and propose new first-order and second-order algorithms. In the latter case, these are the first second-order…
The martingale method is used to establish concentration inequalities for a class of dependent random sequences on a countable state space, with the constants in the inequalities expressed in terms of certain mixing coefficients. Along the…
Random Matrix Theory is a powerful tool in applied mathematics. Three canonical models of random matrix distributions are the Gaussian Orthogonal, Unitary and Symplectic Ensembles. For matrix ensembles defined on k-fold tensor products of…
Following the derivation of the trace formulae in the first paper in this series, we establish here a connection between the spectral statistics of random regular graphs and the predictions of Random Matrix Theory (RMT). This follows from…
We consider three matrix models of order 2 with one random entry $\epsilon$ and the other three entries being deterministic. In the first model, we let $\epsilon\sim\textrm{Bernoulli}\left(\frac{1}{2}\right)$. For this model we develop a…
We investigate the combinatorial discrepancy of geometric set systems having bounded shallow cell complexity in the \emph{Beck-Fiala} setting, where each point belongs to at most $t$ ranges. For set systems with shallow cell complexity…
The evaluation of the error to be attributed to cut efficiencies is a common question in the practice of experimental particle physics. Specifically, the need to evaluate the efficiency of the cuts for background removal, when they are…
We prove existence and uniqueness of the reflected backward stochastic differential equation's (RBSDE) solution with a lower obstacle which is assumed to be right upper-semicontinuous but not necessarily right-continuous in a filtration…
Let $b(x)$ be the probability that a sum of independent Bernoulli random variables with parameters $p_1, p_2, p_3, \ldots \in [0,1)$ equals $x$, where $\lambda := p_1 + p_2 + p_3 + \cdots$ is finite. We prove two inequalities for the…
Many matching, tracking, sorting, and ranking problems require probabilistic reasoning about possible permutations, a set that grows factorially with dimension. Combinatorial optimization algorithms may enable efficient point estimation,…
A classical problem of statistical inference is the valid specification of a model that can account for the statistical dependencies between observations when the true structure is dense, intractable, or unknown. To address this problem, a…
This article develops an analytical framework for studying information divergences and likelihood ratios associated with Poisson processes and point patterns on general measurable spaces. The main results include explicit analytical…
We present two sharp, closed-form empirical Bernstein inequalities for symmetric random matrices with bounded eigenvalues. By sharp, we mean that both inequalities adapt to the unknown variance in a tight manner: the deviation captured by…
We propose a second-order accurate method to estimate the eigenvectors of extremely large matrices thereby addressing a problem of relevance to statisticians working in the analysis of very large datasets. More specifically, we show that…
Number theorists have studied extensively the connections between the distribution of zeros of the Riemann $\zeta$-function, and of some generalizations, with the statistics of the eigenvalues of large random matrices. It is interesting to…
It is well known that bosons and fermions exhibit opposite behaviors when experiencing interference, in the sense that bosons have a tendency to bunch whereas fermions have a tendency to antibunch. Recently, this complementarity was…
We consider large random matrices with a general slowly decaying correlation among its entries. We prove universality of the local eigenvalue statistics and optimal local laws for the resolvent away from the spectral edges, generalizing the…