Second class particles and limit shapes of evacuation and sliding paths for random tableaux

TL;DR

This paper studies the asymptotic behavior of second class particles in TASEP systems and the typical paths in random Young tableaux, revealing convergence to geometric limit shapes like ellipses and meridians.

Contribution

It establishes the limit shape phenomena for second class particles and sliding paths in large random tableaux, connecting probabilistic models to geometric structures.

Findings

Second class particle trajectories converge to elliptical arcs.

Sliding and evacuation paths in large tableaux converge to random meridians.

Results extend to non-square Young tableaux shapes.

Abstract

We investigate two closely related setups. In the first one we consider a TASEP-style system of particles with specified initial and final configurations. The probability of each history of the system is assumed to be equal. We show that the rescaled trajectory of the \emph{second class particle} converges (as the size of the system tends to infinity) to a random arc of an ellipse. In the second setup we consider a uniformly random Young tableau of square shape and look for typical (in the sense of probability) sliding paths and evacuation paths in the asymptotic setting as the size of the square tends to infinity. We show that the probability distribution of such paths converges to a random meridian connecting the opposite corners of the square. We also discuss analogous results for non-square Young tableaux.