Shift-Invariance of the Colored TASEP and Finishing Times of the Oriented Swap Process

TL;DR

This paper establishes a multi-time shift-invariance property for the colored TASEP, linking it to last-passage percolation and providing new asymptotic results for the process and related models.

Contribution

It generalizes existing single-time shift-invariance to multiple times for the colored TASEP, connecting it to last-passage percolation and proving a conjectured distributional identity.

Findings

Proves multi-time shift-invariance of colored TASEP.

Establishes a distributional identity between swap process times and last-passage percolation.

Derives new asymptotic results for the models.

Abstract

We prove a new shift-invariance property of the colored TASEP. From the shift-invariance of the colored stochastic six-vertex model (proved in Borodin-Gorin-Wheeler or Galashin), one can get a shift-invariance property of the colored TASEP at one time, and our result generalizes this to multiple times. Our proof takes the single-time shift-invariance as an input, and uses analyticity of the probability functions and induction arguments. We apply our shift-invariance to prove a distributional identity between the finishing times of the oriented swap process and the point-to-line passage times in exponential last-passage percolation, which is conjectured by Bisi-Cunden-Gibbons-Romik and Bufetov-Gorin-Romik, and is also equivalent to a purely combinatorial identity related to the Edelman-Greene correspondence. With known results from last-passage percolation, we also get new asymptotic results on the colored TASEP and the finishing times of the oriented swap process.