Continuous Multi-line Queues and TASEP

TL;DR

This paper investigates the distribution of labeled particles on a continuous ring related to multi-type TASEP, providing formulas, conjectures, and connections involving Vandermonde determinants.

Contribution

It introduces new formulas and conjectures for particle distributions in continuous multi-line queues linked to TASEP, including probability calculations and structural insights.

Findings

Formulas for permutation probability densities

Conjectures on particle adjacency probabilities

Identified equal probabilities expressed as Vandermonde determinants

Abstract

In this paper, we study a distribution of labeled particles on a continuous ring. It arises in three different ways, all related to the multi-type TASEP on a ring. We prove formulas for the probability density function for some permutations and give conjectures for a larger class. We give a complete conjecture for the probability of two particles i, j being next to each other on the cycle, for which we prove some cases. We also find that two natural events associated to the process have exactly the same probability expressed as a Vandermonde determinant. It is unclear whether this is just a coincidence or a consequence of a deeper connection.