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

We introduce a new interacting particle system on Z, slowed t-TASEP. It may be viewed as a q-TASEP with additional position-dependent slowing of jump rates, depending on a parameter t, which leads to discrete asymptotic fluctuations at large time. If on the other hand t → 1 as time → ∞, we prove 1. A law of large numbers for particle positions, 2. A central limit theorem, with convergence to the fixed-time Gaussian marginal of a stationary solution to SDEs derived from the particle jump rates, and 3. A bulk limit to a certain explicit stationary Gaussian process on R, with scaling exponents characteristic of the Edwards-Wilkinson universality class in (1 + 1)dimensions.The proofs relate slowed t-TASEP to a certain Hall-Littlewood process, and use contour integral formulas for observables of this process. Unlike most previously studied Macdonald processes, this one involves only local interactions, resulting in asymptotics characteristic of (1 + 1)-dimensional rather than (2 + 1)-dimensional systems.

In recent literature, moonshine has been explored for some groups beyond the Monster, for example the sporadic O'Nan and Thompson groups. This collection of examples may suggest that moonshine is a rare phenomenon, but a fundamental and largely unexplored question is how general the correspondence is between modular forms and finite groups. For every finite group G, we give constructions of infinitely many graded infinite-dimensional C[G]-modules where the McKay-Thompson series for a conjugacy class [g] is a weakly holomorphic modular function properly on Γ 0 (ord(g)). As there are only finitely many normalized Hauptmoduln, groups whose McKay-Thompson series are normalized Hauptmoduln are rare, but not as rare as one might naively expect. We give bounds on the powers of primes dividing the order of groups which have normalized Hauptmoduln of level ord(g) as the graded trace functions for any conjugacy class [g], and completely classify the finite abelian groups with this property. In particular, these include (Z/5Z) 5 and (Z/7Z) 4 , which are not subgroups of the Monster.

No 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 [23] and Cuenca [15] 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 and Qiu [12] and Assiotis [1] 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 $A_1, A_2A_1, A_3A_2A_1,\ldots $ of independent Haar-distributed matrices $A_i$ over ${\mathbb {Z}}_p$, generalizing the explicit formula for the classical Cohen–Lenstra measure.