We study the classical two-dimensional one-component plasma of N positively charged point particles, interacting via the Coulomb potential and confined by an external potential. For the specific inverse temperature β = 1 (in our normalization), the charges are the eigenvalues of random normal matrices, and the model is exactly solvable as a determinantal point process. For any positive temperature, using a multiscale scheme of iterated mean-field bounds, we prove that the equilibrium measure provides the local particle density down to the optimal scale of N o(1) particles. Using this result and the loop equation, we further prove that the particle configurations are rigid, in the sense that the fluctuations of smooth linear statistics on any scale are N o(1) .
For the two-dimensional one-component Coulomb plasma, we derive an asymptotic expansion of the free energy up to order N , the number of particles of the gas, with an effective error bound N 1−κ for some constant κ > 0. This expansion is based on approximating the Coulomb gas by a quasi-free Yukawa gas. Further, we prove that the fluctuations of the linear statistics are given by a Gaussian free field at any positive temperature. Our proof of this central limit theorem uses a loop equation for the Coulomb gas, the free energy asymptotics, and rigidity bounds on the local density fluctuations of the Coulomb gas, which we obtained in a previous paper. P G N,V,β (dz) = 1 Z G N,V,β e −βH G N,V (z) m ⊗N (dz), (1.3)
We study one-dimensional exact scaling lognormal multiplicative chaos measures at criticality. Our main results are the determination of the exact asymptotics of the right tail of the distribution of the total mass of the measure, and an almost sure upper bound for the modulus of continuity of the cumulative distribution function of the measure. We also find an almost sure lower bound for the increments of the measure almost everywhere with respect to the measure itself, strong enough to show that the measure is supported on a set of Hausdorff dimension 0.
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