Surjectivity of random integral matrices on integral vectors

Seminar: Probability Seminar | March 11 | 3:10-4 p.m. | 330 Evans Hall

 Melanie Matchett Wood, U.C. Berkeley

 Department of Statistics

A random n by m matrix gives a random linear transformation
from \Z^m to \Z^n (between vectors with integral coordinates). Asking for the probability that such a map is injective is a question of the non-vanishing of determinants. In this talk, we discuss the
probability that such a map is surjective, which is a more subtle
integral question. We show that when m=n+u, for u at least 1, as n
goes to infinity, the surjectivity probability is a non-zero product
of inverse values of the Riemann zeta function. This probability is
universal, i.e. we prove that it does not depend on the distribution
from which you choose independent entries of the matrix. This talk is
on joint work with Hoi Nguyen.