Asymptotic normality of the number of corners in tableaux associated with the partially asymmetric simple exclusion process

Abstract

In this paper, we study corners in tree‐like and permutation tableaux. Tree‐like tableaux are in bijection with other combinatorial structures, including permutation tableaux, and have a connection to the partially asymmetric simple exclusion process (PASEP), an important model of an interacting particles system. In particular, in the context of tree‐like tableaux, a corner corresponds to a node occupied by a particle that could jump to the right while inner corners indicate a particle with an empty node to its left. Thus, the total number of corners represents the number of nodes at which PASEP can move, that is, the total current activity of the system. As the number of inner corners and regular corners is connected, we limit our discussion to just regular corners and show that asymptotically, the number of corners in a tableau of length n is normally distributed. Furthermore, since the number of corners in tree‐like tableaux is closely related to the number of corners in permutation tableaux, we will discuss the corners in the context of the latter tableaux. Finally, using analogous techniques, we prove a central limit theorem for the number of corners in symmetric tree‐like tableaux and type‐B permutation tableaux.