The Asymptotic Distribution of Symbols on Diagonals of Random Weighted Staircase Tableaux

TL;DR

This paper proves that the distribution of symbols on the kth diagonal of random staircase tableaux is asymptotically Poisson with parameter 1/2, and these symbols are asymptotically independent, confirming a conjecture for fixed k.

Contribution

It extends previous results by proving the asymptotic Poisson distribution and independence of symbols on the kth diagonal of staircase tableaux for fixed k.

Findings

Symbols on the kth diagonal follow an asymptotic Poisson distribution with parameter 1/2.

Symbols on the kth diagonal are asymptotically independent.

The conjecture for the distribution of symbols on diagonals beyond the third is confirmed.

Abstract

Staircase tableaux are combinatorial objects that were first introduced due to a connection with the asymmetric simple exclusion process (ASEP) and Askey-Wilson polynomials. Since their introduction, staircase tableaux have been the object of study in many recent papers. Relevant to this paper, the distri- bution of parameters on the first diagonal was proven to be asymptotically normal. In that same paper, a conjecture was made that the other diagonals would be asymptotically Poisson. Since then, only the second and the third diagonal were proven to follow the conjecture. This paper builds upon those results to prove the conjecture for fixed k. In particular, we prove that the distribution of the number of alphas (betas) on the kth diagonal, k > 1, is asymptotically Poisson with parameter 1\2. In addition, we prove that symbols on the kth diagonal are asymptotically independent and thus, collectively follow the Poisson distribution with parameter 1.