Hoi Nguyen

The Ohio State University | OSU · Department of Mathematics

In this paper we study the cokernels of various random integral matrix models, including random symmetric, random skew-symmetric, and random Laplacian matrices. We provide a systematic method to establish universality under very general randomness assumption. Our highlights include both local and global universality of the cokernel statistics of al...

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We prove that the adjacency matrices of uniform random r-regular graphs on n vertices are asymptotically almost surely nonsingular for even n. We use recent mixing results of M\'esz\'aros for adjacency matrices of certain random r-regular multi-graphs and results of the second author on the moment problem for random finite abelian groups.

In this short note we study a non-degeneration property of eigenvectors of symmetric random matrices with entries of symmetric sub-Gaussian distributions. Our result is asymptotically optimal under the sub-exponential regime.

For a random matrix of entries sampled independently from a fairly general distribution in Z we study the probability that the cokernel is isomorphic to a given finite abelian group, or when it is cyclic. This includes the probability that the linear map between the integer lattices given by the matrix is surjective. We show that these statistics a...

We address overcrowding estimates for the singular values of random iid matrices, as well as for the eigenvalues of random Wigner matrices. We show evidence of long range separation under arbitrary perturbation even in matrices of discrete entry distributions. In many cases our method yields nearly optimal bounds.

We show that a nearly square iid random integral matrix is surjective over the integral lattice with very high probability. This answers a question by Koplewitz. Our result extends to sparse matrices as well as to matrices of dependent entries.

We address repulsion property among the singular values of random iid matrices, as well as among the eigenvalues of random Wigner matrices. We show evidence of repulsion under arbitrary perturbation even in matrices of discrete entry distributions. In many cases our method yields nearly optimal bounds in long range repulsion.

It is well-known that distances in random iid matrices are highly concentrated around their mean. In this note we extend this concentration phenomenon to Wigner matrices. Exponential bounds for the lower tail are also included.

We show that permanents of doubly stochastic matrices with balanced entries are not far away from the minimum $n!/n^n$. As an application, we give a general law of large permanent, answering a question by Bochi, Iommi and Ponce

Suppose that A_1,\dots, A_N are independent random matrices whose atoms are iid copies of a random variable \xi of mean zero and variance one. It is known from the works of Newman et. al. in the late 80s that when \xi is gaussian then N^{-1} \log ||A_N \dots A_1|| converges to a non-random limit. We extend this result to more general matrices with...

Let v_1,...,v_{n-1} be n-1 independent vectors in R^n (or C^n). We study x, the unit normal vector of the hyperplane spanned by the v_i. Our main finding is that x resembles a random vector chosen uniformly from the unit sphere, under some randomness assumption on the v_i. Our result has applications in random matrix theory. Consider an n by n rand...