On the asymptotic distribution of parameters in random weighted staircase tableaux

TL;DR

This paper investigates the asymptotic distribution of parameters in random staircase tableaux, providing partial proofs for the conjecture that these distributions are Poisson along certain diagonals as the tableau size grows.

Contribution

It advances understanding by proving the Poisson distribution conjecture for the second and third diagonals of staircase tableaux.

Findings

Distribution along the second diagonal is asymptotically Poisson.

Distribution along the third diagonal is asymptotically Poisson.

Partial proof of the conjecture for higher diagonals.

Abstract

In this paper, we study staircase tableaux, a combinatorial object introduced due to its connections with the asymmetric exclusion process (ASEP) and Askey-Wilson polynomials. Due to their interesting connections, staircase tableaux have been the object of study in many recent papers. More specific to this paper, the distribution of various parameters in random staircase tableaux has been studied. There have been interesting results on parameters along the main diagonal, however, no such results have appeared for other diagonals. It was conjectured that the distribution of the number of symbols along the kth diagonal is asymptotically Poisson as k and the size of the tableau tend to infinity. We partially prove this conjecture; more specifically we prove it for the second and the third main diagonal.