Ian Jauslin

A criterion for crystallization in hard-core lattice particle systems

Computational and Probabilistic Methods: From Theory to Practice, 2025 SIAM New York-New Jersey-Pennsylvania Section Conference, Pennsylvania State University, State College, Pennsylvania, USA

November 1, 2025

As is well known, many materials freeze at low temperatures. Microscopically, this means that their molecules form a phase where there is long range order in their positions. Despite their ubiq- uity, proving that these freezing transitions occur in realistic microscopic models has been a significant challenge, and it remains an open problem in continuum models at positive temperatures. In this talk, I will focus on lattice particle models, in which the positions of particles are discrete. In contrast to the continuum case, there are many examples of lattice particle systems for which crystallization has been proved (in particular in an impressive series of papers by Mazel, Stuhl and Suhov, which cover hard discs on various lattices). In this talk, I will present a general criterion under which crystallization can be proved to occur. The class of models that the criterion applies to are those in which there is no sliding, that is, particles are largely locked in place when the density is large. The tool used in the proof is Pirogov-Sinai theory and cluster expansions. I will present the criterion in its general formulation, and discuss some concrete examples. This is joint work with Qidong He and Joel L. Lebowitz.

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