Rigidity of eigenvalues for β ensemble in multi-cut regime

Abstract

For a β ensemble on ${\mathrm{\Sigma }}^{(\mathit{N})}=\{({\mathit{x}}_1,\dots ,{\mathit{x}}_{\mathit{N}})\in {\mathbb{R}}^{\mathit{N}}|{\mathit{x}}_1\le \cdots \le {\mathit{x}}_{\mathit{N}}\}$ with real analytic potential and general $\mathit{\beta }>0$, under the assumption that its equilibrium measure is supported on q intervals where $\mathit{q}>1$, we prove the following rigidity property for its particles.

1. In the bulk of the spectrum, with overwhelming probability, the distance between a particle and its classical position is of order $\mathit{O}({\mathit{N}}^{-1\mathbf{+}\mathit{ϵ}})$.

2. If k is close to 1 or close to N, that is, near the extreme edges of the spectrum, then with overwhelming probability, the distance between the kth largest particle and its classical position is of order $\mathit{O}({\mathit{N}}^{-\frac{2}{3}\mathbf{+}\mathit{ϵ}}min{(\mathit{k},\mathit{N}\mathbf{+}1-\mathit{k})}^{-\frac{1}{3}})$.

Here $\mathit{ϵ}>0$ is an arbitrarily small constant. Our main idea is to decompose the multi-cut β ensemble as a product of probability measures on spaces with lower dimensions and show that each of these measures is very close to a β ensemble in one-cut regime for which the rigidity of particles is known.