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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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