Continuous Multi-line Queues and the Tasep

In this paper, we study a distribution Υ of labeled particles {1, 2, . . . , n} 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 (cylic) 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 end with observations of the similarities between the TASEP on a ring and the well-studied Razumov-Stroganov process.