Tridiagonal Random Matrix: Gaussian Fluctuations and Deviations

Abstract

This paper is devoted to the Gaussian fluctuations and deviations of the traces of tridiagonal random matrices. Under quite general assumptions, we prove that the traces are approximately normally distributed. A Multi-dimensional central limit theorem is also obtained here. These results have several applications to various physical models and random matrix models, such as the Anderson model, the random birth–death Markov kernel, the random birth–death Q matrix and the $\beta$-Hermite ensemble. Furthermore, under an independent-and-identically-distributed condition, we also prove the large deviation principle as well as the moderate deviation principle for the traces.

Appendix

Proof of (3.13)

By Assumption (H.1) and (H.2),

$$\begin{aligned} \sup \limits _{i\ge 1} {\mathbb {E}}|n^{-{\alpha }k} \widetilde{Q}_{l,i}^{\overrightarrow{m}_l, \overrightarrow{n}_l}|^2 < {\infty }, \end{aligned}$$

(6.1)

where $\widetilde{Q}_{l,i}^{\overrightarrow{m}_l} = Q_{l,i}^{\overrightarrow{m}_l} - {\mathbb {E}}Q_{l,i}^{\overrightarrow{m}_l}$. Hence,

$$\begin{aligned}&\Vert n^{-{\alpha }k} (Tr \widetilde{Q_n^k} - \sum \limits _{i=1}^n \widetilde{X}_{k,i}) \Vert _2 \\&\quad =\Vert \sum \limits _{(l,\overrightarrow{m}_l,\overrightarrow{n}_l)\in \Psi _k} C_l^{\overrightarrow{m}_l, \overrightarrow{n}_l} \sum \limits _{i=n-l+1}^n n^{-{\alpha }k} \widetilde{Q}_{l,i}^{\overrightarrow{m}_l, \overrightarrow{n}_l}\Vert _2 \\&\quad \le \sum \limits _{(l,\overrightarrow{m}_l,\overrightarrow{n}_l)\in \Psi _k} C_l^{\overrightarrow{m}_l, \overrightarrow{n}_l} \sum \limits _{i=n-l+1}^n \Vert n^{-{\alpha }k} \widetilde{Q}_{l,i}^{\overrightarrow{m}_l, \overrightarrow{n}_l}\Vert _2 =\mathcal {O}(1), \end{aligned}$$

with $\Vert \cdot \Vert _2$ denoting the standard $L^2$ norm, which implies (3.13). $\square$

Proof of (3.21)

It is equivalent to prove that, as $n\rightarrow {\infty }$,

$$\begin{aligned} n^{-2{\alpha }k} Cov (X_{k,n}, X_{k,n+j}) \rightarrow Cov (Z_{k,1}, Z_{k,1+j}). \end{aligned}$$

(6.2)

First consider the nonsymmetric case. Let ${\alpha }_i=(a_{i-1}, d_i, b_i)$, $i\ge 1$ and $a_0=0$. By the independence and weak convergence of ${\alpha }_i$ in Assumption (H.1), it is not difficult to deduce that

$$\begin{aligned} n^{-{\alpha }k} ({\alpha }_n, \ldots , {\alpha }_{n+[\frac{k}{2}]}) \overset{d}{\rightarrow } (\widetilde{{\alpha }}_1, \ldots , \widetilde{{\alpha }}_{1+[\frac{k}{2}]}), \end{aligned}$$

where $\widetilde{{\alpha }}_i$ are independent but with the same distribution as that of $(a+ \eta , d+\zeta , b+ \xi )$. Then, by (4.5) and the continuous mapping theorem ([19, Theorem 3.2.4]),

$$\begin{aligned} n^{-{\alpha }k} X_{k,n} \overset{d}{\rightarrow } F(\widetilde{{\alpha }}_1, \ldots , \widetilde{{\alpha }}_{1+[\frac{k}{2}]}) \end{aligned}$$

(6.3)

with F the continuous function defined as in (4.2).

Similarly,

$$\begin{aligned} n^{-{\alpha }k} X_{k,n+j} \overset{d}{\rightarrow } F(\widetilde{{\alpha }}_{1+j}, \ldots , \widetilde{{\alpha }}_{1+j+[\frac{k}{2}]}). \end{aligned}$$

(6.4)

On the other hand, by (3.3) and Hölder’s inequality

$$\begin{aligned} \sup \limits _{n\ge 1} {\mathbb {E}}(n^{-{\alpha }k} X_{k,n})^4 < {\infty }, \end{aligned}$$

(6.5)

which implies the uniform integrabilities of $n^{-2{\alpha }k} X_{k,n}X_{k,n+j}$, $n^{-{\alpha }k} X_{k,n}$ and $n^{-{\alpha }k} X_{k,n+j}$, $n\ge 1$.

Therefore, by (6.3)–(6.5), we can apply the Skorohod representation theorem and the uniform integrability to take the limit and obtain (6.2) for the non-symmetric case. (See, e.g., [19, Theorem 3.2.4] for similar arguments for the bounded continuous mapping.)

The symmetric case can be proved analogously. In fact, with the uniform integrability (6.5), we only need to check the weak convergence of $n^{-{\alpha }k} X_{k,n}$.

In the case that all $d_i$ are independent of $a_i(=b_i)$, let $\beta _i = (d_i, a_i)$, $i\ge 1$. In this case, $X_{k,n}= F'(\beta _n,\ldots , \beta _{n+[\frac{k}{2}]})$ with $F'$ defined as in (4.3). Then, it follows from similar arguments as above that

$$\begin{aligned} n^{-{\alpha }k} X_{k,n} \overset{d}{\rightarrow } F'(\widetilde{\beta _1}, \ldots , \widetilde{\beta }_{1+[\frac{k}{2}]}), \end{aligned}$$

(6.6)

where $\widetilde{\beta _i}$, $1\le i\le [\frac{k}{2}]+1$, are independent but with the common distribution as that of $(d+\xi , a+\eta )$.

In the case that $d_i=f(a_{i-1}, a_i)$, then $\beta _i = (f(a_{i-1}, a_i), a_i)$, $X_{k,n}$ is now a continuous function of $a_{n-1}, \ldots , a_{n+[\frac{k}{2}]}$. As $(a_{n-1},\ldots , a_{n+[\frac{k}{2}]}) \overset{d}{\rightarrow } (\widetilde{{\alpha }_1}, \ldots , \widetilde{{\alpha }}_{[\frac{k}{2}]+2})$, by the continuous mapping theorem, we can obtain the weak convergence of $n^{-{\alpha }k} X_{k,n}$. The proof is consequently complete. $\square$

Proof of (4.4)

Set $N(k)= |\Psi _k|$, the number of sets in $\Psi _k$ which is defined as in (2.3). N(k) is finite and depends only on k. Note that

$$\begin{aligned}&{\mathbb {P}}\left( \frac{1}{n} \left| Tr Q_n^k - \sum \limits _{i=1}^n X_i \right| \ge \delta \right) \\&\quad \le \sum \limits _{(l,\overrightarrow{m}_l,\overrightarrow{n}_l)\in \Psi _k} {\mathbb {P}}\left( \left| C_{l}^{\overrightarrow{m}_l,\overrightarrow{n}_l} \sum \limits _{i=n-l+1}^n Q_{l,i}^{\overrightarrow{m}_l,\overrightarrow{n}_l} \right| \ge \frac{n \delta }{ N(k)} \right) . \end{aligned}$$

Then, letting C denote the maximum of $C_{l}^{\overrightarrow{m}_l,\overrightarrow{n}_l}$ over the finite sets in $\Psi _k$ and setting $\delta _{k} = \delta /([\frac{k}{2}] C N(k))$, we deduce that

$$\begin{aligned} {\mathbb {P}}\left( \frac{1}{n} \left| Tr Q_n^k - \sum \limits _{i=1}^n X_{k,i} \right| \ge \delta \right) \le \sum \limits _{(l,\overrightarrow{m}_l,\overrightarrow{n}_l)\in \Psi _k} \sum \limits _{i=n-l+1}^n {\mathbb {P}}\left( \left| Q_{l,i}^{\overrightarrow{m}_l,\overrightarrow{n}_l} \right| \ge n \delta _{k} \right) . \end{aligned}$$

Since by (2.1) and Assumption (H.3), $\{ |Q _{l,i}^{\overrightarrow{m}_l,\overrightarrow{n}_l}| \}_{i\ge 1}$ is uniformly bounded, which implies that ${\mathbb {P}}(|Q _{l,i}^{\overrightarrow{m}_l,\overrightarrow{n}_l}| \ge n\delta _k) =0$ for n large enough. Hence, by Lemma 1.2.15 in [13],

$$\begin{aligned}&\frac{1}{n} \log \mathbb {P} \left( \frac{1}{n} |Tr Q_n^k - \sum \limits _{i=1}^n X_{k,i}| \ge \delta \right) \nonumber \\&\quad \le \max _{\begin{array}{c} (l,\overrightarrow{m}_l,\overrightarrow{n}_l)\in \Psi _k \\ 1\le j\le [\frac{k}{2}] \end{array}} \frac{1}{n} \log {\mathbb {P}}\left( |Q_{l, n-l+j}^{\overrightarrow{m}_l,\overrightarrow{n}_l}| \ge n \delta _{k} \right) =-{\infty }. \end{aligned}$$

(6.7)

yielding (4.4) as claimed. $\square$

Proof of (4.6)

Similarly to the proof of (6.7), we derive that

$$\begin{aligned}&{\lambda }_n \log \mathbb {P} \left( \sqrt{\frac{{\lambda }_n}{n}} |Tr \widetilde{Q_n^k} - \sum \limits _{i=1}^n \widetilde{X}_{k,i}| \ge \delta \right) \\&\quad \le \max _{\begin{array}{c} (l,\overrightarrow{m}_l,\overrightarrow{n}_l)\in \Psi _k \\ 1\le j\le [\frac{k}{2}] \end{array}} {\lambda }_n \log {\mathbb {P}}\left( |\widetilde{Q}_{l, n-l+j}^{\overrightarrow{m}_l,\overrightarrow{n}_l}| \ge \sqrt{\frac{n}{{\lambda }_n}} \delta _{k} \right) \end{aligned}$$

with $\delta _k$ defined as in the proof of (6.7), which yields (4.6), due to the fact that $\{Q_{l, i}^{\overrightarrow{m}_l,\overrightarrow{n}_l}\}_{i\ge 1}$ is uniformly bounded and $n/{\lambda }_n \rightarrow {\infty }$.