Random surface growth and Karlin-McGregor polynomials

TL;DR

This paper explores the connections between non-intersecting birth and death chains, orthogonal polynomials, and random growth processes, providing new probabilistic proofs and introducing a novel inhomogeneous growth model with explicit correlation kernel calculations.

Contribution

It introduces a general inhomogeneous random growth process with a wall, preserving determinantal structure and enabling explicit correlation kernel computation, linking multiple areas in stochastic processes and representation theory.

Findings

Unified probabilistic proofs of intertwining relations

Explicit correlation kernel for the growth process

Connection to orthogonal polynomials and classical groups

Abstract

We consider consistent dynamics for non-intersecting birth and death chains, originating from dualities of stochastic coalescing flows and one dimensional orthogonal polynomials. As corollaries, we obtain unified and simple probabilistic proofs of certain key intertwining relations between multivariate Markov chains on the levels of some branching graphs. Special cases include the dynamics on the Gelfand-Tsetlin graph considered by Borodin and Olshanski and the ones on the BC-type graph recently studied by Cuenca. Moreover, we introduce a general inhomogeneous random growth process with a wall that includes as special cases the ones considered by Borodin and Kuan and Cerenzia, that are related to the representation theory of classical groups and also the Jacobi growth process more recently studied by Cerenzia and Kuan. Its most important feature is that, this process retains the determinantal structure of the ones studied previously and for the fully packed initial condition we are able to calculate its correlation kernel explicitly in terms of a contour integral involving orthogonal polynomials. At a certain scaling limit, at a finite distance from the wall, one obtains for a single level discrete determinantal ensembles associated to continuous orthogonal polynomials, that were recently introduced by Borodin and Olshanski, and that depend on the inhomogeneities.