Abstract

We consider a Markov chain on nonnegative integer arrays of a given shape (and satisfying certain constraints) which is closely related to fundamental $\mathrm{SL}<!--FUNCTION\; APPLICATION-->(r+1,ℝ)$ Whittaker functions and the Toda lattice. In the index zero case the arrays are reverse plane partitions. We show that this Markov chain has nontrivial 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 semidiscrete 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.

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