Limits and fluctuations of p-adic random matrix products
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Author(s)
Van Peski, Roger
Date Issued
October 2021
Journal
Selecta Mathematica
Publisher
Springer International Publishing
Citation
Van Peski, R. Limits and fluctuations of p-adic random matrix products. Sel. Math. New Ser. 27, 98 (2021)
Version
Author's final manuscript
Abstract
We show that singular numbers (also known as elementary divisors, invariant factors or Smith normal forms) of products and corners of random matrices over Qp are governed by the Hall–Littlewood polynomials, in a structurally identical manner to the known relations between singular values of complex random matrices and Heckman–Opdam hypergeometric functions. This implies that the singular numbers of a product of corners of Haar-distributed elements of GLN(Zp) form a discrete-time Markov chain distributed as a Hall–Littlewood process, with the number of matrices in the product playing the role of time. We give an exact sampling algorithm for the Hall–Littlewood processes which arise by relating them to an interacting particle system similar to PushTASEP. By analyzing the asymptotic behavior of this particle system, we show that the singular numbers of such products obey a law of large numbers and their fluctuations converge dynamically to independent Brownian motions. In the limit of large matrix size, we also show that the analogues of the Lyapunov exponents for matrix products have universal limits within this class of GLN(Zp) corners.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1007/s00029-021-00709-3
