Poisson Statistics for Matrix Ensembles at Large Temperature

Abstract

In this article, we consider $\beta$-ensembles, i.e. collections of particles with random positions on the real line having joint distribution

$$\begin{aligned} \frac{1}{Z_N(\beta )}\big |\Delta (\lambda )\big |^\beta e^{- \frac{N\beta }{4}\sum _{i=1}^N\lambda _i^2}\mathrm {d}\lambda , \end{aligned}$$

in the regime where $\beta \rightarrow 0$ as $N\rightarrow \infty$. We briefly describe the global regime and then consider the local regime. In the case where $N\beta$ stays bounded, we prove that the local eigenvalue statistics, in the vicinity of any real number, are asymptotically to those of a Poisson point process. In the case where $N\beta \rightarrow \infty$, we prove a partial result in this direction.

Appendix: Poisson Limit for Point Processes

Let $\mathcal {X}$ be a locally compact Polish space and $\mu$ be a Radon measure on $\mathcal {X}$. We consider an exchangeable random vector $(\lambda _1, \ldots , \lambda _N)$ taking values on $\mathcal {X}$ implicitly depending on N, with density $\rho ^{(N)}$ with respect to $\mu ^{\otimes N}$. We define, for $1\le k\le N$, the k-th correlation function on $\mathcal {X}^k$ by the formula

$$\begin{aligned} R^{(N)}_k(x_{1}, \ldots , x_k):=\frac{N!}{(N-k)!}\int _{(x_{k+1},\ldots , x_N)\in \mathcal {X}^{N-k} }\rho ^{(N)}(x_1, \ldots , x_N)\mathrm {d}\mu ^{\otimes N-k}(x_{k+1},\ldots , x_N). \end{aligned}$$

Proposition 5.6

Suppose that there is $\theta \ge 0$ independent of N such that the correlation functions $R_k^{(N)}$ satisfy:

• (a) For each $k\ge 1$, on $\mathcal {X}^k$, we have the pointwise convergence

  $$\begin{aligned} R^{(N)}_k(x_{1}, \ldots , x_k)\;\underset{N\rightarrow \infty }{\longrightarrow }\;\theta ^k, \end{aligned}$$

  (63)

• (b) For each compact $\mathcal {K}\subset \mathcal {X}$, there is $\Theta _{\mathcal {K}}$ such that for all k, N, on $\mathcal {K}^k$, we have

  $$\begin{aligned} \mathbbm {1}_{k\le N}R^{(N)}_k(x_{1}, \ldots , x_k) \;\le \;\Theta _{\mathcal {K}}^k \end{aligned}$$

  (64)

Then the point process $\sum _{i=1}^N\delta _{\lambda _i}$ converges in distribution to a Poisson point process with intensity $\theta \mathrm {d}\mu$ as $N\rightarrow \infty$.

Proof

Note that the Poisson point process M with intensity $\theta \mathrm {d}\mu$ is characterized, among random Radon measures on $\mathcal {X}$, by the fact that for any compactly supported continuous function f on $\mathcal {X}$, we have

$$\begin{aligned} {\mathbb {E}}\mathrm {e}^{\langle M, f\rangle }=\exp \left(\theta \int \left( \mathrm {e}^{f(x)}-1\right) \mathrm {d}\mu (x)\right). \end{aligned}$$

So let us fix f a compactly supported continuous function on $\mathcal {X}$. Then, with the convention $R_0^{(N)}=1$,

$$\begin{aligned} {\mathbb {E}}\mathrm {e}^{\sum _{i=1}^Nf(\lambda _i)}= & {} {\mathbb {E}}\prod _{i=1}^N\left( 1+(\mathrm {e}^{f(\lambda _i)}-1)\right) \\= & {} \sum _{P\subset \{1, \ldots , n\}}{\mathbb {E}}\prod _{i\in P}\left( \mathrm {e}^{f(\lambda _i)}-1\right) \\= & {} \sum _{k=0}^N\left( {\begin{array}{c}N\\ k\end{array}}\right) {\mathbb {E}}\prod _{i=1}^k\left( \mathrm {e}^{f(\lambda _i)}-1\right) \\= & {} \sum _{k=0}^N\frac{1}{k!}\int \prod _{i=1}^k\left( \mathrm {e}^{f(x_i)}-1\right) R^{(N)}_k(x_{1}, \ldots , x_k)\mathrm {d}\mu ^{\otimes k}(x_1, \ldots , x_k) \end{aligned}$$

This proves the proposition. $\square$