Limiting empirical spectral distribution conjecture for randomly permuted sparse matrices

Fix $\alpha \in [0,1)$. Let $A=PLQ$ be an $n\times n$ random matrix, where $P$ and $Q$ are independent uniformly chosen permutation matrices and $L$ is independent of $P$ and $Q$. The nonzero entries of $L$ are independent copies of a random variable $\xi$ satisfying $\mathbb E\xi=0$, $\mathbb E|\xi|=\sigma^2$, and $\mathbb E|\xi|^4<\infty$, and $L_{ij}=0$ if $i+\lfloor k_\alpha\rfloor<j$. Write $\mu_{A_n/\sqrt{n\sigma^2(1-\alpha)}}$ for the empirical spectral distribution of the scaled matrix, and let $\mathbb D$ denote the unit disk, $\nu$ the uniform probability measure on $\mathbb D$, and $\delta_0$ the Dirac measure at the origin. The limiting spectral distribution conjecture. There exists a probability measure $\nu_\alpha$ on $\mathbb D$ such that $\mu_{A_n/\sqrt{n\sigma^2(1-\alpha)}}$ converges weakly to $\nu_\alpha$ in probability and almost surely. Moreover, $\nu_\alpha\to\nu$ as $\alpha\to0$ and $\nu_\alpha\to\delta_0$ as $\alpha\to1$, with both convergences holding uniformly with respect to total variation distance. The conjectured family $\nu_\alpha$ interpolates between the uniform measure on the unit disk and the Dirac measure at the origin. Empirical evidence motivates this conjecture as a universality class beyond the circular law for fixed sparsity; the limiting measure is expected to depend on $\alpha$ but not on the distribution of $\xi$.