A CLT for the characteristic polynomial of random Jacobi matrices, and the G$\beta$E

Abstract

We prove a central limit theorem for the real part of the logarithm of the characteristic polynomial of random Jacobi matrices. Our results cover the G$\beta$E models for $\beta >0$.

Appendix A: An anti-concentration inequality

The Lévy anti-concentration function $Q_\mu (\cdot )$ of a probability measure $\mu \in {\mathcal {P}}({\mathbb {R}})$ is defined as

$$\begin{aligned} Q_\mu (\varepsilon ) = \sup _{ x \in {\mathbb {R}}} \mu \big ([x-\varepsilon , x+\varepsilon ]\big ), \qquad \varepsilon >0. \end{aligned}$$

(A.1)

For a random variable $X\sim \mu$, we often write $Q_X$ for $Q_\mu$.

Lemma A.1

For $M\ge 1$, set

$$\begin{aligned} {\mathcal {C}}_M=\left\{ \mu \in {\mathcal {P}}({\mathbb {R}}): \int x d\mu =0, \quad \int x^2 d\mu =1, \quad \int |x|^3 d\mu \le M\right\} . \end{aligned}$$

(A.2)

Then for any $\varepsilon \in (0,1/4)$ there exists $\delta =\delta (M,\varepsilon )>0$ such that for all $\mu \in {\mathcal {C}}_M$,

$$\begin{aligned} Q_\mu (\varepsilon ) \le 1-\delta . \end{aligned}$$

(A.3)

Proof

Let $\varepsilon >0$, and assume by contradiction that there exists a sequence $\mu _n \in {\mathcal {C}}_M$ such that $Q_{\mu _n}(\varepsilon ) \underset{n\rightarrow \infty }{\longrightarrow }\ 1.$ As ${\mathcal {C}}_M$ is a compact set for the weak convergence, there exist a sequence $n_k$ and $\mu \in {\mathcal {P}}({\mathbb {R}})$ such that $\mu _{n_k} \underset{k\rightarrow \infty }{\leadsto }\mu .$ Since $\int |x|^3 d\mu _{n_k} \le M$, we deduce by Fatou’s Lemma that $\mu \in {\mathcal {C}}_M$. Observe that as $Q_{\mu _n}(\varepsilon ) \rightarrow 1$ when $n\rightarrow \infty$, we obtain using Markov’s inequality that for n large enough,

$$\begin{aligned} Q_{\mu _n}(\varepsilon ) = \sup _{|x| \le 2M+\varepsilon }\mu _n\big ( [x-\varepsilon ,x+\varepsilon ]\big ). \end{aligned}$$

Now, let $\phi _\varepsilon$ be a non-negative Lipschitz function such that $\phi _{\varepsilon } =1$ on $[-\varepsilon ,\varepsilon ]$ and $\phi _\varepsilon =0$ on $[-2\varepsilon ,2\varepsilon ]^c$, and define for any probability measure $\mu$,

$$\begin{aligned} \tilde{Q}_{\mu }(\varepsilon ) = \sup _{|x| \le 2M+\varepsilon } \int \phi _\varepsilon (x-y) d\mu . \end{aligned}$$

By continuity, we deduce that for any k there exists $|x_k|\le 2\,M +\varepsilon$ such that

$$\begin{aligned} \tilde{Q}_{\mu _{n_k}}(\varepsilon ) = \int \phi _\varepsilon (x_k-y) d\mu _{n_k}. \end{aligned}$$

Since $Q_{\mu _{n_k}}(\varepsilon )\le \tilde{Q}_{\mu _{n_k}}(\varepsilon )$, we get that $\int \phi _\varepsilon (x_k-y) d\mu _{n_k} \underset{k\rightarrow \infty }{\longrightarrow }\ 1.$ At the price of taking another sub-sequence we can assume that $x_k$ converges to some $x \in {\mathbb {R}}$. We then obtain that $\int \phi _\varepsilon (x-y) d\mu = 1,$ which yields that $\mu \big ([x-2\varepsilon ,x+2\varepsilon ]\big ) =1$. Since $\textrm{Var}(\mu ) =1$ and $(4\varepsilon )^2 <1$, we get a contradiction. $\square$

Combining Lemma A.1 with the Kolmogorov-Rogozin inequality [31], we get the following anti-concentration inequality for sums of independent random variables.

Lemma A.2

Assume that $X_1,X_2,\ldots ,X_m$ are independent centered random variables such that for some $M>0$, ${\mathbb {E}}|X_i|^3 \le M ({\mathbb {E}}X_i^2)^{3/2}$ for $i=1,\ldots ,m$. Let $S = X_1+X_2+\ldots +X_m$. Then there is a constant $C=C(M)$ so that for any $t \ge \max _i \sqrt{{\mathbb {E}}(X_i^2)}/8$,

$$\begin{aligned} Q_S(t) \le \frac{Ct}{\sqrt{ \sum _{i=1}^m {\mathbb {E}}X_i^2}}. \end{aligned}$$

Proof

Note that the statement and its assumptions are scale-invariant, so we may and will assume that $\max _{i=1}^m {\mathbb {E}}X_i^2=1$. By the Kolmogorov-Rogozin inequality [31], we have that for any constants $s_i\le t$,

$$\begin{aligned} Q_S(t) \le \frac{Ct}{\sqrt{\sum _{i=1}^m s_i^2 (1-Q_{X_i}(s_i))}}. \end{aligned}$$

Let $s_i =\sqrt{{\mathbb {E}}(X_i^2)}/8$, $i=1,\ldots ,m$. By Lemma A.1, we have that $Q_{X_i}(s_i)\le 1-\delta$, where $\delta >0$ only depends on M. This completes the proof. $\square$