Discrete Whittaker processes

TL;DR

This paper introduces a Markov chain on integer arrays linked to Whittaker functions and the Toda lattice, revealing new Markovian projections, entrance laws, and connections to various mathematical models.

Contribution

It develops a novel Markov chain model related to Whittaker functions, explores its properties, and establishes connections to multiple integrable systems and probabilistic models.

Findings

Existence of non-trivial Markovian projections.

Unique entrance law from infinite array.

Connections to Brownian motion functionals and polymer models.

Abstract

We consider a Markov chain on non-negative integer arrays of a given shape (and satisfying certain constraints) which is closely related to fundamental $SL(r+1,\mathbb{R})$ Whittaker functions and the Toda lattice. In the index zero case the arrays are reverse plane partitions. We show that this Markov chain has non-trivial Markovian projections and a unique entrance law starting from the array with all entries equal to $+\infty$. We also discuss connections with imaginary exponential functionals of Brownian motion, a semi-discrete polymer model with purely imaginary disorder, interacting corner growth processes and discrete $\delta$-Bose gas, extensions to other root systems, and hitting probabilities for some low rank examples.