On the probability that a discrete complex random matrix is singular

Abstract

OF THE DISSERTATION On the probability that a discrete complex random matrix is singular by Philip J. Wood Dissertation Director: Van H. Vu Let n be a large integer and Mn be an n by n complex matrix whose entries are independent (but not necessarily identically distributed) discrete random variables. The main goal of this thesis is to prove a general upper bound for the probability that Mn 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 Mn satisfy this property, then the probability that Mn is singular is at most ( p1/r + o(1) )n . All of the results in this thesis hold for any characteristic zero integral domain replacing the complex numbers. In the special case where the entries of Mn are “fair coin flips” (taking the values +1,−1 each with probability 1/2), our general bound implies that the probability that Mn 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 [39]. In the special case where the entries of Mn 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 Mn is singular is at most ( 1 2 + o(1) )n , which is asymptotically sharp.