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1D log gases and the renormalized energy: crystallization at vanishing temperature. (English) Zbl 1327.82005
The statistical mechanics of a one-dimensional log gas or beta-ensemble with general potential and arbitrary beta (the inverse of temperature) is studied according to the method introduced by the same authors [Ann. Probab. 43, No. 4, 2026–2083 (2015; Zbl 1328.82006)] for 2D Coulomb gases. The formal limit beta going to infinity corresponds to the “weighted Fekete sets” and it is also treated her. A one-dimensional version of the renormalized energy of the same authors [Commun. Math. Phys. 313, No. 3, 635–743 (2012; Zbl 1252.35034)] is introduced that measures the total logarithmic interaction of an infinite set of points on the real line in a uniform neutralizing background. It is shown that this energy is minimized when the points are on a lattice. By a suitable splitting of the Hamiltonian the full statistical mechanics problem is connected to this renormalized energy and this allows to obtain new results on the distribution of the points at the microscopic scale. In particular, it is shown that configurations whose \(W\) is above a certain threshold (tending to \(\min W\) as \(\beta\) goes to infinity) have exponentially small probability. This shows that such configurations have increasing when the temperature goes to zero.

MSC:
82B05 Classical equilibrium statistical mechanics (general)
82B21 Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics
82B26 Phase transitions (general) in equilibrium statistical mechanics
15B52 Random matrices (algebraic aspects)
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