Discrete Whittaker processes

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.