p-adic random matrices and particle systems

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. Both prelimit and limit objects exhibit much more spatial independence than their analogues for complex matrices, often with surprising results. In different regimes we can prove Gaussian limits, an intriguing new discrete Poisson-type local limit (yielding a local interacting particle system on $\mathbb{Z}$ similar to $q$-TASEP), and convergence of global bulk fluctuations to a certain explicit Gaussian process.

mathematical physicsprobability

Audience: researchers in the topic

Probability and the City Seminar

Series comments: The Probability and the City Seminar is organized jointly by the probability groups of Columbia University and New York University.

Video recordings of talks are posted online at www.youtube.com/channel/UC0CXjG-ZSIZHy0S40Px2FEQ .