Limit shapes for the asymmetric five vertex model

TL;DR

This paper derives explicit limit shapes for the asymmetric five-vertex model by computing free energy and surface tension, applying a variational principle, and solving the resulting Euler-Lagrange equations.

Contribution

It introduces a variational approach and explicit solutions for limit shapes in the five-vertex model, expanding understanding of non-intersecting lattice path models.

Findings

Explicit formulas for free energy and surface tension.

Analytic parameterization of limit shapes.

Euler-Lagrange equations integrated for explicit solutions.

Abstract

We compute the free energy and surface tension function for the five-vertex model, a model of non-intersecting monotone lattice paths on the grid in which each corner gets a positive weight. We give a variational principle for limit shapes in this setting, and show that the resulting Euler-Lagrange equation can be integrated, giving explicit limit shapes parameterized by analytic functions.