Hall-Littlewood polynomials, boundaries, and $p$-adic random matrices

Abstract

We prove that the boundary of the Hall-Littlewood t-deformation of the Gelfand-Tsetlin graph is parametrized by infinite integer signatures, extending results of Gorin [Gor12] and Cuenca [Cue18] on boundaries of related deformed Gelfand-Tsetlin graphs. In the special case when 1/t is a prime p we use this to recover results of Bufetov-Qiu [BQ17] and Assiotis [Ass20] on infinite p-adic random matrices, placing them in the general context of branching graphs derived from symmetric functions.Our methods rely on explicit formulas for certain skew Hall-Littlewood polynomials. As a separate corollary to these, we obtain a simple expression for the joint distribution of the cokernels of products A1, A2A1, A3A2A1, . . . of independent Haar-distributed matrices Ai over the p-adic integers Zp. This expression generalizes the explicit formula for the classical Cohen-Lenstra measure on abelian p-groups. Contents 1. Introduction 2. Hall-Littlewood polynomials 3. Principally specialized skew Hall-Littlewood polynomials 4. The t-deformed Gelfand-Tsetlin graph and its boundary 5. Infinite p-adic random matrices and corners 6. Ergodic decomposition of p-adic Hua measures. 7. Products of finite p-adic random matrices Appendix A. Commuting dynamics and projection to the boundary References

$q$-TASEP with position-dependent slowing

“…However, the n = ∞ case of (1.2), which corresponds to the full slowed t-TASEP, can interpreted as the projection of Hall-Littlewood process dynamics to a row 'at infinity'. A fuller account is given in [VP21a], but the basic idea is that with the initial condition where every entry of the Gelfand-Tsetlin pattern is 0, at each time all sufficiently high rows of the Gelfand-Tsetlin pattern will yield the same partition, and the projection of the dynamics to this partition is Markovian and yields the n = ∞ case of (1.2). For a precise formulation of this statement in terms of the boundary of a branching graph, see [VP21a, Appendix A].…”