Separation in low temperature beta-ensembles

Problem's Description

The following result is known for a repelling system of $N$ particles in a plane, at inverse temperature $\beta$: if we zoom at a suitable point at a natural microscopic scale, and if $\beta$ is at least proportional to $\log N$, then the particles become (almost-certainly) separated from each other by a certain fixed distance, as $N->\infty$.

I suspect that the above result might be "tight" in the following sense: if an almost certain asymptotic separation holds on the microscopic level (say, in the situation of "Corollary" on page 1081) then $\beta$ necessarily increases at least proportional to $\log N$. My question is if someone can prove or refute this statement, or alternatively to provide some new and interesting angle on it.

1. Article Repulsion in low temperature beta-ensembles

   • Yacin Ameur

   fulltext