Local density for two-dimensional one-component plasma

TL;DR

This paper analyzes the local density and rigidity of particles in a two-dimensional one-component plasma, demonstrating precise density estimates and fluctuation bounds across scales using exact solvability and mean-field techniques.

Contribution

It introduces a multiscale scheme to establish local density estimates and rigidity for the 2D one-component plasma at any positive temperature.

Findings

Local particle density is accurately estimated down to near microscopic scales.

Particle configurations exhibit rigidity with fluctuations smaller than any polynomial in N.

The model's exact solvability at inverse temperature 1 facilitates precise analysis.

Abstract

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 $\beta=1$ (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 $N^{o(1)}$.