Asymptotics of Schur functions on almost staircase partitions

TL;DR

This paper investigates the asymptotic behavior of Schur functions for almost staircase partitions, revealing convergence properties and connections to probabilistic models like dimer configurations on lattices.

Contribution

It provides new asymptotic formulas for Schur functions of almost staircase partitions using determinant and integral methods.

Findings

Convergence of scaled log ratios of Schur functions to sum of holomorphic functions

Results linked to law of large numbers for dimer models

Connections to central limit theorem for specific lattice configurations

Abstract

We study the asymptotics of Schur polynomials with partitions $\lambda$ which are almost staircase; more precisely, partitions that differ from $((m-1)(N-1),(m-1)(N-2),\ldots,(m-1),0)$ by at most one component at the beginning as $N\rightarrow \infty$, for a positive integer $m\ge 1$ independent of $N$. By applying either determinant formulas or integral representations for Schur functions, we show that $\frac{1}{N}\log \frac{s_{\lambda}(u_1,\ldots,u_k, x_{k+1},\ldots,x_N)}{s_{\lambda}(x_1,\ldots,x_N)}$ converges to a sum of $k$ single-variable holomorphic functions, each of which depends on the variable $u_i$ for $1\leq i\leq k$, when there are only finitely many distinct $x_i$'s and each $u_i$ is in a neighborhood of $x_i$, as $N\rightarrow\infty$. The results are related to the law of large numbers and central limit theorem for the dimer configurations on contracting square-hexagon lattices with certain boundary conditions.