# Topological limit shape phase transitions: melting of Arctic Circles

A limit shape phenomenon in statistical mechanics is the appearance of a most probable macroscopic state. An iconic example of this phenomenon is given by the Arctic Circle Theorem [1] of random tilings which, in the certain scaling limit, can be mapped to the imaginary time evolution of free fermions. A limit shape is usually characterized by a well-defined boundary separating frozen and liquid spatial regions. The earliest studies related to this phenomenon in the context of crystal shapes

are in works by Pokrovsky and Talapov [2].

In this talk, I will present a phase transition of limit shape, which can be visualized as merging two melted regions (Arctic circles). By mapping onto a free fermionic problem and calculating correspondent correlation functions we identify the transition as the third-order transition known in lattice QCD [3]. We make connections to algebraic geometry, stressing the topological nature of the transition and identifying universal features of the limiting shape.

[1] W. Jockusch, J. Propp and P. Shor, arXiv preprint math/9801068 (1998).

"Random domino tilings and the arctic circle theorem."

[2] V. L. Pokrovsky and A. L. Talapov, Phys. Rev. Lett. 42, 65 (1979).

"Ground State, Spectrum, and Phase Diagram of Two-Dimensional Incommensurate

Crystals."

[3] D. J. Gross and E. Witten, Phys. Rev. D, 21 (2): 446, 1980. "Possible

third-order phase transition in the large-n lattice gauge theory.”;

S. R. Wadia, "N = ∞ phase transition in a class of exactly soluble model

lattice gauge theories", Phys. Lett. 93, 403 (1980)

## Department of Physics

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