Grove arctic curves from periodic cluster modular transformations

Terrence George

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arXiv · 1711.00790

Grove arctic curves from periodic cluster modular transformations

Terrence George

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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.

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Terrence George. 2017-11-02. Grove arctic curves from periodic cluster modular transformations. https://arxiv.org/abs/1711.00790

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