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.
”Macdonald processes.” Probability Theory and Related Fields 158.1 (2014): 225-400
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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.