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