Infinite 𝑝-adic random matrices and ergodic decomposition of 𝑝-adic Hua measures

Abstract

Neretin in [Izv. Ross. Akad. Nauk Ser. Mat. 77 (2013), pp. 95–108] constructed an analogue of the Hua measures on the infinite p p -adic matrices M a t ( N , Q p ) \mathrm {Mat}\left (\mathbb {N},\mathbb {Q}_p\right ) . Bufetov and Qiu in [Compos. Math. 153 (2017), pp. 2482–2533] classified the ergodic measures on M a t ( N , Q p ) \mathrm {Mat}\left (\mathbb {N},\mathbb {Q}_p\right ) that are invariant under the natural action of G L ( ∞ , Z p ) × G L ( ∞ , Z p ) \mathrm {GL}(\infty ,\mathbb {Z}_p)\times \mathrm {GL}(\infty ,\mathbb {Z}_p) . In this paper we solve the problem of ergodic decomposition for the p p -adic Hua measures introduced by Neretin. We prove that the probability measure governing the ergodic decomposition has an explicit expression which identifies it with a Hall-Littlewood measure on partitions. Our arguments involve certain Markov chains.

Hall–Littlewood Polynomials, Boundaries, and p-Adic Random Matrices

“…A key difference which will be relevant later is that Sig N is a discrete set. Some previous works in p-adic random matrix theory by Neretin [54], Bufetov-Qiu [19], Assiotis [5], and the author [63,64], which come from a more Lie-theoretic standpoint than those mentioned above, have found close structural analogies between singular values of complex matrices and their analogues for p-adic matrices. This begged the question of whether the multiplicative Brownian motion on GL N (C) and multiplicative Dyson Brownian motion have analogues in the p-adic setting.…”

“…We note that previous works[5,19,54] use the opposite sign convention on singular numbers; we use the above one so that they are nonnegative when A ∈ MatN (Zp) 3. This uniqueness actually applies after restricting to the subgroup SLN (C).…”