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We study the statistical mechanics of a one-dimensional log gas with generalpotential and arbitrary beta, the inverse of temperature, according to themethod we introduced for two-dimensional Coulomb gases in [SS2]. Such ensemblescorrespond to random matrix models in some particular cases. The formal limitbeta infinite corresponds to "weighted Fekete sets" and is also treated. Weintroduce a one-dimensional version of the "renormalized energy" of [SS1],measuring the total logarithmic interaction of an infinite set of points on thereal line in a uniform neutralizing background. We show that this energy isminimized when the points are on a lattice. By a suitable splitting of theHamiltonian we connect the full statistical mechanics problem to thisrenormalized energy W, and this allows us to obtain new results on thedistribution of the points at the microscopic scale: in particular we show thatconfigurations whose W is above a certain threshhold (which tends to min W asbeta tends to infinity) have exponentially small probability. This shows thatthe configurations have increasing order and crystallize as the temperaturegoes to zero.

1D Log Gases and the Renormalized Energy: Crystallization at Vanishing Temperature