Grove arctic curves from periodic cluster modular transformations

TL;DR

This paper introduces a broad class of probability measures on groves, enabling explicit computation of arctic curves that generalize the arctic circle theorem, revealing complex phase structures in the limit shapes.

Contribution

It develops a method to compute arctic curves for groves using asymptotics of multivariate generating functions, extending previous results to more general measures.

Findings

Explicit arctic curves for a wide class of grove measures

Limit shapes exhibit all expected solid and gaseous phases

Generalization of the arctic circle theorem to new settings

Abstract

Groves are spanning forests of a finite region of the triangular lattice that are in bijection with Laurent monomials that arise in solutions of the cube recurrence. We introduce a large class of probability measures on groves for which we can compute exact generating functions for edge probabilities. Using the machinery of asymptotics of multivariate generating functions, this lets us explicitly compute arctic curves, generalizing the arctic circle theorem of Petersen and Speyer. Our class of probability measures is sufficiently general that the limit shapes exhibit all solid and gaseous phases expected from the classification of EGMs in the resistor network model.