Asymptotics of Schur functions on almost staircase partitions

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.