Independent GUE minor processes of perfect matchings on rail-yard graphs

TL;DR

This paper investigates perfect matchings on rail-yard graphs with specific boundary conditions and demonstrates that certain dimer distributions near the boundary converge to independent GUE minor process spectra, using advanced Schur function analysis.

Contribution

It introduces a new analysis of Schur functions to prove convergence of dimer distributions to independent GUE minor processes in a novel graph setting.

Findings

Dimer distributions near the boundary converge to GUE minor spectra.

Established a new quantitative Schur function analysis method.

Extended understanding of perfect matchings on rail-yard graphs.

Abstract

We study perfect matchings on the rail-yard graphs in which the right boundary condition is given by the empty partition and the left boundary can be divided into finitely many alternating line segments where all the vertices along each line segment are either removed or remained. When the edge weights satisfy certain conditions, we show that the distributions of the locations of certain types of dimers near the right boundary converge to the spectra of independent GUE minor processes. The proof is based on new quantitative analysis of a formula to compute Schur functions at general points discovered in \cite{ZL18}.