ÍâÍøÌìÌÃ

As is well known, many materials freeze at low temperatures. Microscopically,Ìý this means that their molecules form a phase where there is long range orderÌý in their positions. Despite their ubiquity, proving that these freezingÌý transitions occur in realistic microscopic models has been a significantÌý challenge, and it remains an open problem in continuum models at positiveÌýtemperatures. In this talk, I will focus on lattice particle models, in whichÌý the positions of particles are discrete, and discuss a general criterionÌýunder which crystallization can be proved to occur. The class of models thatÌý the criterion applies to are those in which there is *no sliding*, that is,Ìýparticles are largely locked in place when the density is large. The toolÌý used in the proof is Pirogov-Sinai theory and cluster expansions. I willÌýpresent the criterion in its general formulation, and discuss some concreteÌý examples. This is joint work with Qidong He and Joel L. Lebowitz.