Spectrum of finite-band random matrices?

Question

Let $X_n=(X_{ij})_{1 \leq i,j \leq n}$ such that :

$$ \begin{cases} &X_{ij} = 0 \quad \text{if}\quad \vert i - j \vert > k\\ & X_{ij} \sim P_X \quad \text{otherwise} \end{cases}$$

And the non zero entries are iid, $k \in \mathbb{N}$ is a $\textbf{fixed}$ integer. $X_n$ is a band random matrix of size k indepedent of n.

Note that $X_n$ is not hermitian.

Now I know that when $k = k(n) \gg \sqrt n $ one recovers the circular law after proper renormalization.

However, in that case, one needs not renormalize (the energy of the matrix is finite).

Denote by $\mu_{X_n}$ the empirical spectral measure. Do we have : $$\mu_{X_n} \underset{n \to \infty}\longrightarrow \mu_{\infty}^X \quad \text{?}$$

Note : It is clear that if the limit exist, it depends on the law $P_X$ of $X$, indeed in the case $k=1$ the matrix is diagonal, and it is easy to see that

$$k=1 \implies \mu_{X_n} \underset{n \to \infty}\longrightarrow P_X $$

Maybe the limit is known for special cases ? Like the Gaussian one maybe ?

Answer 1

Yes, convergence to a limiting empirical measure $\mu_\infty$ is well known . Let $L\in \mathbb{N}$, (we will chose $1\ll L \ll n$) and define $Y$ $$Y_{i,j}=\begin{cases}X_{i,j} \text{ if } \exists k\in \mathbb{N} \text{ such that } i,j \in [kL,(k+1)L-1] \\ 0 \text{ otherwise} \end{cases} $$ Then $Y$ is a diagonal block matrix of independent block and each blok are iid. Therefore its spectral measure is the sum of the spectral measure of each block which are IID random variables. So $\mu_Y$ converge to $\mu_L$ the mean of the spectral measure of a block.

Now we use that $$d(\mu_X,\mu_Y)\leq \frac{1}{n}\|Y-X\|_1\sim \frac{k}{L}$$ and denoting $\epsilon = k/L$, we obtain that $\mu_X$ is a Cauchy sequence.

But there is no known formula to calculte the limiting measure. The semi circular law should appear as soon as $k\rightarrow \infty$ (stronger than the result you state) .

Answer 2

Band random matrices with a band width $k$ that does not increase with the size $n$ of the matrix are representations of the random Schrödinger equation. The simplest example is a matrix of the form $$H=\begin{pmatrix} V_1&1&0&0&\cdots&0\\ 1&V_2&1&0&\cdots&0\\ 0&1&V_3&1&\cdots&0\\ .&.&.&.&.&.\\ 0&0&0&0&1&V_n \end{pmatrix}$$ This is a discretized version of the Schrödinger equation $$-\psi''(x)+V(x)\psi(x)=E\psi(x)$$ on a one-dimensional lattice with a random potential $V$. The eigenvalue statistics is Poissonian, there is no level repulsion because of localization.