Limit shape of perfect matchings on rail-yard graphs

TL;DR

This paper derives the limit shape and frozen boundary equations for perfect matchings on rail-yard graphs with various boundary conditions, identifying when the boundary forms cloud curves or unions thereof.

Contribution

It introduces a comprehensive analysis of limit shapes for perfect matchings on rail-yard graphs with new boundary conditions and characterizes the frozen boundary as cloud curves or unions.

Findings

Explicit parametric equations for the frozen boundary.

Conditions for the frozen boundary to be a cloud curve.

Identification of union of disjoint cloud curves as frozen boundaries.

Abstract

We obtain limit shape of perfect matchings on a large class of rail-yard graphs with right boundary condition given by the empty partition, and left boundary condition given by either by a staircase partition with constant density or a piecewise partition with densities either 1 or 0. We prove the parametric equations for the frozen boundary, and find conditions under which the frozen boundary is a cloud curve, or a union of disjoint cloud curves.