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

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 A₁, A₂A₁, A₃A₂A₁,… of independent Haar-distributed matrices Aᵢ over Zₚ, generalizing the explicit formula for the classical Cohen–Lenstra measure.