Nearest neighbor Markov dynamics on Macdonald processes

TL;DR

This paper classifies all continuous-time, nearest neighbor Markov dynamics acting on Macdonald processes, unifying known examples and discovering new integrable particle systems and combinatorial correspondences.

Contribution

It provides a complete classification of such dynamics on Macdonald processes, including new integrable models and combinatorial structures, extending previous work.

Findings

Unified known dynamics under a classification framework

Discovered a new q-deformed PushTASEP particle system

Identified new Robinson--Schensted-type correspondences

Abstract

Macdonald processes are certain probability measures on two-dimensional arrays of interlacing particles introduced by Borodin and Corwin (arXiv:1111.4408 [math.PR]). They are defined in terms of nonnegative specializations of the Macdonald symmetric functions and depend on two parameters (q,t), where 0<= q, t < 1. Our main result is a classification of continuous time, nearest neighbor Markov dynamics on the space of interlacing arrays that act nicely on Macdonald processes. The classification unites known examples of such dynamics and also yields many new ones. When t = 0, one dynamics leads to a new integrable interacting particle system on the one-dimensional lattice, which is a q-deformation of the PushTASEP (= long-range TASEP). When q = t, the Macdonald processes become the Schur processes of Okounkov and Reshetikhin (arXiv:math/0107056 [math.CO]). In this degeneration, we discover new Robinson--Schensted-type correspondences between words and pairs of Young tableaux that govern some of our dynamics.