We consider the probability measures on Young diagrams in the \(n\times k\) rectangle obtained by specializations with parameters given by piecewise continuously differentiable functions of Schur polynomials in the dual Cauchy identity. We derive the uniform convergence to a limit shape of Young diagrams in the limit \(n,k\rightarrow \infty \) . 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. 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. 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 et al. (J Stat Phys 102(5–6):1085–1132, 2001) restricting to the leading order. The general nature of our parameters provides strong evidence of universality of the Airy, Pearcey and discrete Hermite kernels at the edge, in analogy with recent results in random matrix theory.
