Emergence of a Poisson process in weakly interacting particle systems

TL;DR

This paper demonstrates that in weakly interacting particle systems, the local point process converges to a Poisson process within a specific temperature range, expanding the known conditions for such convergence.

Contribution

It extends the temperature regime under which convergence to a Poisson point process is proven for weakly interacting particle systems.

Findings

Convergence to Poisson process occurs when $N^{-1} \\ll \\beta \\ll N^{-1/2}$.

Expands the temperature range for proven Poisson convergence.

Applicable to general interacting particle systems with Gibbs measure.

Abstract

We consider the Gibbs measure of a general interacting particle system for a certain class of ``weakly interacting" kernels. In particular, we show that the local point process converges to a Poisson point process as long as the inverse temperature $\beta$ satisfies $N^{-1} \ll \beta \ll N^{-\frac{1}{2}}$, where $N$ is the number of particles. This expands the temperature regime for which convergence to a Poisson point process has been proved.