Analysis of moments and cumulants in alternating sign matrices

TL;DR

This paper derives exact formulas and asymptotic expansions for moments and cumulants of certain observables in alternating sign matrices, revealing deep combinatorial and analytic structures.

Contribution

It introduces new exact formulas and asymptotic analysis for moments and cumulants in alternating sign matrices using integrable kernel methods.

Findings

Exact formulas for expectations using Bernoulli polynomials

Asymptotic expansions involving Riemann zeta functions

Cumulants up to fourth order analyzed

Abstract

In this work, we study the discrete observables $$E_k = \sum_{i,j=1}^n (i-j)^k A_{i,j}$$ associated with $n\times n$ alternating sign matrices $A = (A_{i,j})$. This work develops exact formulas for expectations using Bernoulli polynomials, exponential generating functions, expansions in $1/n$ linked to Riemann zeta functions, and cumulants up to fourth order via integrable kernel methods. All intermediate calculations, expansions, and pedagogical details are provided to illustrate the interplay between combinatorial sums, analytic expansions, and integrable structures in alternating sign matrices.