English

On the singularity probability of discrete random matrices

Combinatorics 2009-05-05 v1

Abstract

Let MnM_n be an nn by nn random matrix where each entry is +1 or -1 independently with probability 1/2. Our main result implies that the probability that MnM_n is singular is at most (1/2+o(1))n(1/\sqrt{2} + o(1))^n, improving on the previous best upper bound of (3/4+o(1))n(3/4 + o(1))^n proven by Tao and Vu in arXiv:math/0501313v2. This paper follows a similar approach to the Tao and Vu result, including using a variant of their structure theorem. We also extend this type of exponential upper bound on the probability that a random matrix is singular to a large class of discrete random matrices taking values in the complex numbers, where the entries are independent but are not necessarily identically distributed.

Keywords

Cite

@article{arxiv.0905.0461,
  title  = {On the singularity probability of discrete random matrices},
  author = {Jean Bourgain and Van Vu and Philip Matchett Wood},
  journal= {arXiv preprint arXiv:0905.0461},
  year   = {2009}
}

Comments

45 pages, two figures

R2 v1 2026-06-21T12:58:04.055Z