2D Coulomb Gases and the Renormalized Energy

摘要: We study the statistical mechanics of classical two-dimensional “Coulomb gases” with general potential and arbitrary $\beta$, inverse temperature. Such ensembles also correspond to random matrix models in some particular cases. The formal limit case $\beta=\infty$ corresponds “weighted Fekete sets” falls within our analysis.It is known that such a system points should be asymptotically distributed according macroscopic “equilibrium measure,” large deviations principle holds for this, as proven by Petz Hiai [In Advances Differential Equations Mathematical Physics (Atlanta, GA, 1997) (1998) Amer. Math. Soc.] Ben Arous Zeitouni [ ESAIM Probab. Statist.2 123–134]. By suitable splitting Hamiltonian, we connect problem “renormalized energy” $W$, Coulombian interaction plane introduced [ Comm. Phys.313 (2012) 635–743], which expected good way measuring disorder an infinite configuration plane. so doing, are able examine situation at microscopic scale, obtain several new results: next order asymptotic expansion partition function, estimates on probability fluctuation from equilibrium measure microscale, type result, states configurations above certain threshhold $W$ have exponentially small probability. When $\beta\to\infty$, estimate becomes sharp, showing has “crystallize” minimizer $W$. In weighted sets, this saying these sets microscopically look almost everywhere like minimizers conjectured “Abrikosov” triangular lattices.