Discrete and continuous Muttalib–Borodin processes: The hard edge

Abstract

In this note, we study a natural measure on plane partitions giving rise to a certain discrete-time Muttalib–Borodin process (MBP): each time slice is a discrete version of a Muttalib–Borodin ensemble (MBE). The process is determinantal with explicit-time-dependent correlation kernel. Moreover, in the $q\to 1$ limit, it converges to a continuous Jacobi-like MBP with Muttalib–Borodin marginals supported on the unit interval. This continuous process is also determinantal with explicit correlation kernel. We study its hard-edge scaling limit (around 0) to obtain a discrete-time-dependent generalization of the classical continuous Bessel kernel of random matrix theory (and, in fact, of the Meijer $G$-kernel as well). We lastly discuss two related applications: random sampling from such processes and their interpretations as models of directed last passage percolation (LPP). In doing so, we introduce a corner growth model naturally associated to Jacobi processes, a version of which is the “usual” corner growth of Forrester–Rains in logarithmic coordinates.