For every q, there is a Markov process on reverse plane partitions whose projection to any skew shape is again a Markov process, giving a q-deformed analogue of O'Connell's discrete Whittaker processes.
”On q-Deformed-Whittaker Function I.” Communications in Mathematical Physics 294.1 (2010): 97-119
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$q$-Deformed Discrete Whittaker Processes
For every q, there is a Markov process on reverse plane partitions whose projection to any skew shape is again a Markov process, giving a q-deformed analogue of O'Connell's discrete Whittaker processes.