The Crystallization Conjecture: A Review

Abstract

In this article we describe the crystallization conjecture. It states that, in appropriate physical conditions, interacting particles always place themselves into periodic configurations, breaking thereby the natural translation-invariance of the system. This famous problem is still largely open. Mathematically, it amounts to studying the minima of a real-valued function defined on R 3N where N is the number of particles, which tends to infinity. We review the existing literature and mention several related open problems, of which many have not been thoroughly studied.

On Percolation of Two-Dimensional Hard Disks

“…In this section we prove some corollaries of the Main Theorem related to the notion of a Gibbs distribution. These results are direct counterparts of [1,Theorems 2,3].…”

“…We also menton that there are numerous papers considering similar question for another models of statistical mechanics. See, for example [3,9,16].…”

Large deviation principle for empirical fields of Log and Riesz gases

“…• It is conjectured that the triangular (or Abrikosov) lattice has minimal energy in the Log2 case (see [BL15] for a survey). Can we at least prove that any minimizer of the renormalized energy has infinite specific relative entropy, which would be a first hint towards their conjectural "ordered" nature?…”

Multiscale semidefinite programming approach to positioning problems with pairwise structure

“…For this problem, we have a strong intuition that the support of the optimal 1-marginal is nearly a subset of a hexagonal lattice [3]. Therefore we remove the degeneracy with respect to rigid motions by fixing the first three 'anchor' particles as vertices of an equilateral triangle in the center of the search domain.…”