BBP Transition for Deformed Inhomogeneous Random Matrices

We study finite-rank deformations of inhomogeneous random symmetric/Hermitian matrices $H_N=({\sigma }_{ij}{W}_{ij}{)}_{i,j=1}^N$, whose variance profile satisfies ${\sum }_j{\sigma }_{ij}^2=1$. Writing ${\sigma }_N^{\ast }={\max }_{i,j}{\sigma }_{ij}$ for the matrix sparsity parameter, we establish the BBP transition of outlier eigenvalues at the level of the law of large numbers in two regimes of moment conditions on $\{W_{ij}\}$: (1) under a uniform sub-Weibull bound with exponent $0<\alpha \le 2$ and the sharp assumption ${\sigma }_N^{\ast }(\log N{)}^{1\mathrm{/}\alpha }\to 0$, the convergence holds almost surely; (2) under uniformly integrable $p$-th powers for some $p\ge 4$ and the sharp assumption $N({\sigma }_N^{\ast }{)}^{p-2}=O(1)$, the convergence holds in probability. Our approach combines sharp norm estimates, fixed-degree polynomial isotropic laws, and a finite-dimensional resolvent equation.