  We consider a classical system of n charged particles in an external
confining potential, in any dimension d larger than 2. The particles interact
via pairwise repulsive Coulomb forces and the coupling parameter scales like
the inverse of n (mean-field scaling). By a suitable splitting of the
Hamiltonian, we extract the next to leading order term in the ground state
energy, beyond the mean-field limit. We show that this next order term, which
characterizes the fluctuations of the system, is governed by a new
"renormalized energy" functional providing a way to compute the total Coulomb
energy of a jellium (i.e. an infinite set of point charges screened by a
uniform neutralizing background), in any dimension. The renormalization that
cuts out the infinite part of the energy is achieved by smearing out the point
charges at a small scale, as in Onsager's lemma. We obtain consequences for the
statistical mechanics of the Coulomb gas: next to leading order asymptotic
expansion of the free energy or partition function, characterizations of the
Gibbs measures, estimates on the local charge fluctuations and factorization
estimates for reduced densities. This extends results of Sandier and Serfaty to
dimension higher than two by an alternative approach.
