Six-vertex model with rare corners and random restricted permutations

TL;DR

This paper investigates the limit shapes in the six-vertex model at a specific limit and in restricted Mallows permutations, deriving equations and solutions for these shapes in polygonal domains.

Contribution

It provides the Euler-Lagrange equation for the limit shape and explicit solutions for certain polygonal domains in these models.

Findings

Derived the Euler-Lagrange equation for the limit shape.

Solved the equation explicitly for rectilinear polygonal domains.

Found solutions are piecewise-algebraic functions with discontinuities.

Abstract

We study limit shapes in two equivalent models: the six-vertex model in the $c\to0$ limit and the random Mallows permutation with restricted permutation matrix. We give the Euler-Lagrange equation for the limit shape and show how to solve it for a class of rectilinear polygonal domains. Its solutions are given by piecewise-algebraic functions with lines of discontinuities.