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.

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

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

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 ?