Limits and fluctuations of p-adic random matrix products

Abstract

We study the distribution of singular numbers of products of certain classes of padic random matrices, as both the matrix size and number of products go to ∞ simultaneously. In this limit, we prove convergence of the local statistics to a new random point configuration on Z, defined explicitly in terms of certain intricate mixed q-series/exponential sums. This object may be viewed as a nontrivial p-adic analogue of the interpolating distributions of Akemann-Burda-Kieburg [6], which generalize the sine and Airy kernels and govern limits of complex matrix products. Our proof uses new Macdonald process computations and holds for matrices with iid additive Haar entries, corners of Haar matrices from GLN (Zp), and the p-adic analogue of Dyson Brownian motion studied in [80]. Contents 1. Introduction 1 2. p-adic matrix background 3. Symmetric function background 4. The limit of the Plancherel/principal Hall-Littlewood measure 5. Residue expansions 6. Tightness and the limiting random variable 7. An indeterminate moment problem 8. Examples of residue formula for L k,t,χ 9. The case of pure α specializations 10. From processes of infinitely many particles to finite matrix bulk limits Appendix A. Parallels with complex matrix products Appendix B. A Hall-Littlewood proof of [80, Theorem 1.2] References

q-TASEP with position-dependent slowing

“…, t n−1 . We mention also that this locality, for a slightly different Hall-Littlewood process, was previously exploited in [VP21] in the context of p-adic random matrix theory.…”

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.…”

“…Similar results relating matrix products to multiplicative Dyson Brownian motion in the complex setting have been shown by Ahn [1] (see also [2]). See [62] for more detail on the analogy between these results in the complex and p-adic settings.…”

“…Our motivation to highlight S (N ) in this paper also comes from a related work [62] where its reflected Poisson dynamics are shown to govern the local dynamics of singular numbers of p-adic matrix products under weak universality hypotheses. Similar results relating matrix products to multiplicative Dyson Brownian motion in the complex setting have been shown by Ahn [1] (see also [2]).…”

On the distribution of equivalence classes of random symmetricp‐adic matrices

“…In our case, we may also observe that random p ‐adic matrices are intimately connected with Hall–Littlewood polynomials, which are a generalisation of Schur polynomials used in the work of Diaconis and Shahshahani. Optimistically, we may even expect that with appropriate normalisation results over p ‐adics should morph into the Euclidean case as $$p \rightarrow \infty$$, see, for example, [35].…”