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 A(1), A(2)A(1), A(3)A(2)A(1), ... of independent Haar-distributed matrices A(i) over Z(p), generalizing the explicit formula for the classical Cohen-Lenstra measure.