We consider a Markov chain on nonnegative integer arrays of a given shape
(and satisfying certain constraints) which is closely related to fundamental
SL(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
+∞.
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
δ-Bose
gas, extensions to other root systems, and hitting probabilities for some low rank
examples.
Keywords
Whittaker functions, Toda lattice, reverse plane partitions
