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 $((m-1)(N-1),(m-1)(N-2),\dots ,(m-1),0)$ by at most one component at the beginning as $N\to \infty$, for a positive integer $m\ge 1$ independent of $N$. By applying either determinant formulas or integral representations for Schur functions, we show that $\frac{1}{N}\log \frac{s_{\lambda }(u_1,\dots ,u_k,x_{k+1},\dots ,x_N)}{s_{\lambda }(x_1,\dots ,x_N)}$ converges to a sum of $k$ single-variable holomorphic functions, each of which depends on the variable $u_i$ for $1\le i\le k$, when there are only finitely many distinct $x_i$'s and each $u_i$ is in a neighborhood of $x_i$, as $N\to \infty$. 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.

References in corpus (1)

• Local limits of lozenge tilings are stable under bounded boundary height perturbations