Limit shapes and fluctuations for $(GL_n, GL_k)$ skew Howe duality

TL;DR

This paper analyzes the asymptotic behavior of Young diagrams under certain probability measures related to skew Howe duality, revealing universal limit shapes and fluctuation kernels such as sine, Airy, Hermite, and Pearcey, with connections to random matrix theory.

Contribution

It introduces a comprehensive analysis of limit shapes and fluctuations for Young diagrams in the context of $(GL_n, GL_k)$ skew Howe duality, utilizing free fermionic representations.

Findings

Bulk fluctuations follow the discrete sine kernel.

Boundary fluctuations are described by Tracy-Widom distribution and Airy kernel.

Special boundary sections exhibit Pearcey kernel fluctuations.

Abstract

We consider the probability measures on Young diagrams in the $n \times k$ rectangle obtained by piecewise-continuously differentiable specializations of Schur polynomials in the dual Cauchy identity. We use a free fermionic representation of the correlation kernel to study its asymptotic behavior and derive the uniform convergence to a limit shape of Young diagrams in the limit $n,k \to \infty$. More specifically, we show the bulk is the discrete sine kernel with boundary fluctuations generically given by the Tracy-Widom distribution with the Airy kernel. When our limit shape touches the boundary corner of the rectangle, the fluctuations with a second order correction are given by the discrete Hermite kernel, and we recover the discrete distribution of Gravner-Tracy-Widom (2001) [arXiv:math/0005133] restricting to the leading order. Finally, we demonstrate our limit shapes can have sections with no or full density of particles, where the Pearcey kernel appears when such a section is infinitely small.