Large deviations of crowding in finite $\beta$-ensembles

We consider finite $\beta$-ensembles ${\mathcal{X}}_{n,\beta }^{\mathbb{F}}$ with $n$ points on $\mathbb{F}$, where $\mathbb{F}$ denotes either the real line or the complex plane. Let $U$ be a bounded subset of $\mathbb{F}$ such that $\mathrm{\partial }U$ (the boundary of $U$) is polar for $\mathbb{F}=\mathbb{R}$ and $\mathrm{\partial }U$ is a closed $1$–rectifiable set with finite $1$-dimensional Hausdorff measure. Suppose ${\mathcal{X}}_{n,\beta }^{\mathbb{F}}(U)$ denotes the number of points in the region $U$. We show that the sequence of laws of $\{n^{-1}{\mathcal{X}}_{n,\beta }^{\mathbb{F}}(U);n\ge 1\}$ satisfies the large deviation type bound with speed $n^2$ and with a good rate function. For $\mathbb{F}=\mathbb{R}$, this result can be derived using the contraction principle. However, when $\mathbb{F}=\mathbb{C}$, the contraction principle does not yield the desired outcome. Therefore, we adopt a direct approach to establish our results.