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Elementary divisors and determinants of random matrices over a local field. (English) Zbl 1075.15500
Summary: We consider the elementary divisors and determinant of a uniformly distributed \(n \times n\) 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 \(n \to \infty \), 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 \(M_n\) and find the limit of this distribution as \(n \to \infty \). Our formulae have connections with classical identities for \(q\)-series, and the \(q\)-binomial theorem, in particular.

MSC:
15B52 Random matrices (algebraic aspects)
60B99 Probability theory on algebraic and topological structures
12J25 Non-Archimedean valued fields
05A30 \(q\)-calculus and related topics
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