Multispecies inhomogeneous $t$-PushTASEP with general capacity

TL;DR

This paper introduces an integrable multi-species stochastic process called $t$-PushTASEP with inhomogeneities and arbitrary site capacities, providing explicit stationary probabilities and a novel partition function expression.

Contribution

It develops a comprehensive algebraic framework for the $t$-PushTASEP with inhomogeneities and arbitrary capacities, linking it to quantum group representations and vertex models.

Findings

Explicit stationary probabilities via matrix product form.

Partition function derived from $l=1$ case through plethystic substitution.

Connection established between stochastic process and quantum algebra structures.

Abstract

We study an $n$-species $t$-PushTASEP, an integrable long-range stochastic process, on a one-dimensional periodic lattice with inhomogeneities $x_1,\ldots,x_L$ and arbitrary capacity $l$ at each lattice site. The Markov matrix is identified with an alternating sum of commuting transfer matrices over all fundamental representations of $U_t(\widehat{sl}_{n+1})$. Stationary probabilities are expressed in a matrix product form involving a fusion of quantized corner transfer matrices for the strange five-vertex model introduced by Okado, Scrimshaw, and the second author. The resulting partition function, which serves as the normalization factor of the stationary probabilities, is obtained from the $l=1$ case by a finite plethystic substitution of length $l$.