Correlations in the Multispecies TASEP and a Conjecture by Lam

TL;DR

This paper investigates correlations in the multispecies TASEP on a ring, proving conjectures related to random walks and partitions, and deriving explicit formulas for various correlation types, advancing understanding of the model's structure.

Contribution

It proves two conjectures by Lam regarding asymptotic shapes and directions, and derives explicit formulas for correlations in the multispecies TASEP, extending prior results.

Findings

Proved conjectures on the limiting direction of a reduced random walk and the shape of n-core partitions.

Derived explicit formulas for two-point and three-point correlations in the TASEP.

Identified an independence property between points closer in position than in value.

Abstract

We study correlations in the multispecies TASEP on a ring. Results on correlation of two adjacent points prove two conjectures by Thomas Lam on (a) the limiting direction of a reduced random walk in $\tilde A_{n-1}$ and (b) the asymptotic shape of a random integer partition with no hooks of length $n$, a so called $n$-core. We further investigate two-point correlations far apart and three-point nearest neighbour correlations and prove explicit formulas in almost all cases. These results can be seen as a finite strengthening of correlations in the TASEP speed process by Amir, Angel and Valk\'o. We also give conjectures for certain higher order nearest neighbour correlations. We find an unexplained independence property (provably for two points, conjecturally for more points) between points that are closer in position than in value that deserves more study.