Elementary Divisors and Determinants of Random Matrices over a Local Field

We consider the elementary divisors and determinant of a uniformly distributed n n random matrix with entries in the ring of integers of an arbitrary local eld. We show that the sequence of elementary divisors is in a simple bijective correspondence with a Markov chain on the nonnegative integers. The transition dynamics of this chain do not depend on the size of the matrix. As n ! 1, all but nitely many of the elementary divisors are 1, and the remainder arise from a Markov chain with these same transition dynamics. We also obtain the distribution of the determinant of Mn and nd the limit of this distribution as n ! 1. Our formulae have connections with classical identities for q-series, and the q-binomial theorem in particular. Department of Statistics #3860 University of California at Berkeley 367 Evans Hall Berkeley, CA 94720-3860 U.S.A e-mail: [email protected] URL: http://www.stat.berkeley.edu/users/evans