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p-adic random matrices and particle systems

Combinatorics Seminar
March 30, 2023
10:20AM - 11:15AM
MW 154

Date Range
Add to Calendar 2023-03-30 10:20:00 2023-03-30 11:15:00 p-adic random matrices and particle systems Title:  p-adic random matrices and particle systems Speaker:  Roger Van Peski (MIT) Abstract:  Random p-adic matrices have been studied since the late 1980s as natural models for random groups appearing in number theory and combinatorics. Recently it has also become clear that the theory has close structural parallels with singular values of complex random matrices, bringing new techniques from integrable probability and motivating new questions. After outlining this area (no background in p-adic matrices will be assumed), I will discuss results on the distribution of analogues of singular values for products of many random p-adic matrices. In different regimes we can prove both Gaussian limits and an intriguing new discrete local limit, both appearing quite universally. The latter is an interacting particle system on $\mathbb{Z}$ defined for any real $p > 1$ (analogous to general beta random matrix theory), with an interesting $p \to 1$ limit which sheds light on key differences between p-adic and classical random matrices. The talk will touch on results in https://arxiv.org/abs/2011.09356, https://arxiv.org/abs/2112.03725, and work in preparation. MW 154 Department of Mathematics math@osu.edu America/New_York public

Title:  p-adic random matrices and particle systems

Speaker:  Roger Van Peski (MIT)

Abstract:  Random p-adic matrices have been studied since the late 1980s as natural models for random groups appearing in number theory and combinatorics. Recently it has also become clear that the theory has close structural parallels with singular values of complex random matrices, bringing new techniques from integrable probability and motivating new questions. After outlining this area (no background in p-adic matrices will be assumed), I will discuss results on the distribution of analogues of singular values for products of many random p-adic matrices. In different regimes we can prove both Gaussian limits and an intriguing new discrete local limit, both appearing quite universally. The latter is an interacting particle system on $\mathbb{Z}$ defined for any real $p > 1$ (analogous to general beta random matrix theory), with an interesting $p \to 1$ limit which sheds light on key differences between p-adic and classical random matrices.

The talk will touch on results in https://arxiv.org/abs/2011.09356, https://arxiv.org/abs/2112.03725, and work in preparation.

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