We prove the universality for the eigenvalue gap statistics in the bulk of the spectrum for band matrices, in the regime where the band width is comparable with the dimension of the matrix, $W \sim N$. All previous results concerning universality of non-Gaussian random matrices are for mean-field models. By relying on a new mean-field reduction technique, we deduce universality from quantum unique ergodicity for band matrices.

We develop mathematical models describing the evolution of stochastic age-structured populations. After reviewing existing approaches, we formulate a complete kinetic framework for age-structured interacting populations undergoing birth, death and fission processes in spatially dependent environments. We define the full probability density for the population-size age chart and find results under specific conditions. Connections with more classical models are also explicitly derived. In particular, we show that factorial moments for non-interacting processes are described by a natural generalization of the McKendrick-von Foerster equation, which describes mean-field deterministic behavior. Our approach utilizes mixed-type, multidimensional probability distributions similar to those employed in the study of gas kinetics and with terms that satisfy BBGKY-like equation hierarchies.

Large dimensional Gram type matrices are common objects in high-dimensional statistics and machine learning. In this paper, we study the limiting distribution of the edge eigenvalues for a general class of high-dimensional Gram type random matrices, including separable sample covariance matrices, sparse sample covariance matrices, bipartite stochastic block model and random Gram matrices with general variance profiles. Specifically, we prove that under (almost) sharp moment conditions and certain tractable regularity assumptions, the edge eigenvalues, i.e., the largest few eigenvalues of non-spiked Gram type random matrices or the extremal bulk eigenvalues of spiked Gram type random matrices, satisfy the Tracy-Widom distribution asymptotically. Our results can be used to construct adaptive, accurate and powerful statistics for high-dimensional statistical inference. In particular, we propose data-dependent statistics to infer the number of signals under general noise structure, test the one-sided sphericity of separable matrix, and test the structure of bipartite stochastic block model. Numerical simulations show strong support of our proposed statistics. The core of our proof is to establish the edge universality and Tracy-Widom distribution for a rectangular Dyson Brownian motion with regular initial data. This is a general strategy to study the edge statistics for high-dimensional Gram type random matrices without exploring the specific independence structure of the target matrices. It has potential to be applied to more general random matrices that are beyond the ones considered in this paper.