Deformed Fr\'echet law for Wigner and sample covariance matrices with tail in crossover regime

Given $A_n:=\frac{1}{\sqrt{n}}(a_{ij})$ an $n\times n$ symmetric random matrix, with elements above the diagonal given by i.i.d. random variables having mean zero and unit variance. It is known that when $\lim_{x\to\infty}x^4\mathbb{P}(|a_{ij}|>x)=0$, then fluctuation of the largest eigenvalue of $A_n$ follows a Tracy-Widom distribution. When the law of $a_{ij}$ is regularly varying with index $\alpha\in(0,4)$, then the largest eigenvalue has a Fr\'echet distribution. An intermediate regime is recently uncovered in \cite{diaconu2023more}: when $\lim_{x\to\infty}x^4\mathbb{P}(|a_{ij}|>x)=c\in(0,\infty)$, then the law of the largest eigenvalue follows a deformed Fr\'echet distribution. In this work we vastly extend the scope where the latter distribution may arise. We show that the same deformed Fr\'echet distribution arises (1) for sparse Wigner matrices with an average of $n^{O(1)}$ nonzero entries on each row; (2) for periodically banded Wigner matrices with bandwidth $d_n=n^{O(1)}$; and more generally for weighted adjacency matrices of any $k_n$-regular graphs with $k_n=n^{O(1)}$. In all these cases, we further prove that the joint distribution of the finitely many largest eigenvalues of $A_n$ form a deformed Poisson process, and that eigenvectors of the outlying eigenvalues of $A_n$ are localized, implying a mobility edge phenomenon at the spectral edge $2$. The sparser case with average degree $n^{o(1)}$ is also explored. Our technique extends to sample covariance matrices, proving for the first time that its largest eigenvalue still follows a deformed Fr\'echet distribution, assuming the matrix entries satisfy $\lim_{x\to\infty}x^4\mathbb{P}(|a_{ij}|>x)=c\in(0,\infty)$.