# Separation in low temperature beta-ensembles

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Posted online: 2018-11-19 08:33:35Z by Yacin Ameur31

Cite as: P-181119.3

• Mathematical Physics

### 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.

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