On the singularity probability of discrete random matrices

Abstract

Let n be a large integer and M n be an n by n complex matrix whose entries are independent (but not necessarily identically distributed) discrete random variables. The main goal of this paper is to prove a general upper bound for the probability that M n is singular. For a constant 0 < p < 1 and a constant positive integer r, we will define a property p-bounded of exponent r. Our main result shows that if the entries of M n satisfy this property, then the probability that M n is singular is at most ( p 1 / r + o ( 1 ) ) n . All of the results in this paper hold for any characteristic zero integral domain replacing the complex numbers. In the special case where the entries of M n are “fair coin flips” (taking the values + 1 , − 1 each with probability 1/2), our general bound implies that the probability that M n is singular is at most ( 1 2 + o ( 1 ) ) n , improving on the previous best upper bound of ( 3 4 + o ( 1 ) ) n , proved by Tao and Vu [Terence Tao, Van Vu, On the singularity probability of random Bernoulli matrices, J. Amer. Math. Soc. 20 (2007) 603–628]. In the special case where the entries of M n are “lazy coin flips” (taking values + 1 , − 1 each with probability 1/4 and value 0 with probability 1/2), our general bound implies that the probability that M n is singular is at most ( 1 2 + o ( 1 ) ) n , which is asymptotically sharp. Our method is a refinement of those from [Jeff Kahn, János Komlós, Endre Szemerédi, On the probability that a random ±1-matrix is singular, J. Amer. Math. Soc. 8 (1) (1995) 223–240; Terence Tao, Van Vu, On the singularity probability of random Bernoulli matrices, J. Amer. Math. Soc. 20 (2007) 603–628]. In particular, we make a critical use of the structure theorem from [Terence Tao, Van Vu, On the singularity probability of random Bernoulli matrices, J. Amer. Math. Soc. 20 (2007) 603–628], which was obtained using tools from additive combinatorics.