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

Abstract

Neretin in [38] constructed an analogue of the Hua measures on the infinite p-adic matrices Mat N, Q p . Bufetov and Qiu in [19] classified the ergodic measures on Mat N, Q p that are invariant under the natural action of GL(∞, Z p ) × GL(∞, Z p ). In this paper we solve the problem of ergodic decomposition for the 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

“…Ergodic measures on infinite p-adic random matrices. In the special case t = 1/p, the purely combinatorial results on Hall-Littlewood polynomials have consequences in p-adic random matrix theory, and we may deduce results of [BQ17,Ass20] from Theorem 1.1 above. We refer to Section 7 for basic background on the p-adic integers Z p and p-adic field Q p .…”

“…We now define a special family of measures on Mat ∞×∞ (Q p ), the p-adic Hua measures, introduced in [Ner13]. Their decomposition into the ergodic measures Ẽµ of Definition 19 was computed in [Ass20]. We will rederive that result, showing in the process that the p-adic Hua measures have a natural interpretation in terms of measures on partitions derived from Hall-Littlewood polynomials.…”

“…A motivating property of these measures is that they are consistent under taking corners, and hence define a measure M (s) ∞ on Mat ∞×∞ (Q p ). The decomposition of this measure into ergodic measures on Mat ∞×∞ (Q p ) was computed recently in [Ass20], and we reprove the result using the aforementioned relation between p-adic matrix corners and the Hall-Littlewood branching graph G t . Below E µ is as in Theorem 1.2, Y is the set of integer partitions, Q λ is the dual normalization of the Hall-Littlewood symmetric function, and the normalizing constant Π(1, .…”

“…The key ingredient in the original proof of Theorem 1.3 given previously in [Ass20] is a certain Markov chain which generates the finite Hua measures M (s) n , and which was guessed from Markov chains appearing in similar settings [Ful02]. The arguments there did not use Hall-Littlewood polynomials, but the limiting measure on GT ∞ which describes the ergodic decomposition was observed in [Ass20] to be the so-called Hall-Littlewood measure in (1.5), by matching explicit formulas. From our perspective, by contrast, the fact that this measure is a Hall-Littlewood measure is natural and is key to the proof.…”

“…Acknowledgements. I am grateful to Alexei Borodin for many helpful conversations throughout the project and detailed feedback on several drafts, Theo Assiotis and Alexander Bufetov for comments and suggestions, Grigori Olshanski for discussions on branching graphs, Vadim Gorin for feedback and the suggestion to recover results of [Ass20], and to Nathan Kaplan, Hoi Nguyen, and Melanie Matchett Wood for discussions on cokernels of matrix products. This material is based on work partially supported by an NSF Graduate Research Fellowship under grant #1745302, and by the NSF FRG grant DMS-1664619.…”