Elementary divisors and determinants of random matrices over a local field

Steven N. Evans

Stochastic Processes and their Applications, 2002, vol. 102, issue 1, 89-102

Abstract: We consider the elementary divisors and determinant of a uniformly distributed nxn random matrix with entries in the ring of integers of an arbitrary local field. We show that the sequence of elementary divisors is in a simple bijective correspondence with a Markov chain on the non-negative integers. The transition dynamics of this chain do not depend on the size of the matrix. As n-->[infinity], all but finitely 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 find the limit of this distribution as n-->[infinity]. Our formulae have connections with classical identities for q-series, and the q-binomial theorem, in particular.

Keywords: Local; field; p-Adic; p-Series; Random; matrix; Elementary; divisor; Gaussian; elimination; Determinant; q-Binomial; coefficient; Partition (search for similar items in EconPapers)
Date: 2002
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