2024/08/21 by Dan Betea, Betea, Dan, Anton Nazarov +5 · 1 citation
Mathematics · #05A19 #60C05 #60G55 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Finite Group Theory Research #Mathematical Physics (math-ph) #Probability (math.PR) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2408.11419
openalex publication_date 2024/08/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the probability measures on Young diagrams in the n × 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 → ∞. 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.