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

### Łukasz Maślanka

Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warszawa, Poland### Piotr Śniady

Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warszawa, Poland

## 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 *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.

## Cite this article

Łukasz Maślanka, Piotr Śniady, Second class particles and limit shapes of evacuation and sliding paths for random tableaux.. Doc. Math. 27 (2022), pp. 2183–2273

DOI 10.4171/DM/X28