  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.
