BBP Transition for Deformed Inhomogeneous Random Matrices
Abstract
We study finite-rank deformations of inhomogeneous random symmetric/Hermitian matrices HN=(σijWij)i,j=1N, whose variance profile satisfies ∑jσij2=1. Writing σN*=maxi,jσij 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 {Wij}: (1) under a uniform sub-Weibull bound with exponent 0<α≤ 2 and the sharp assumption σN*(log N)1/α→ 0, the convergence holds almost surely; (2) under uniformly integrable p-th powers for some p≥ 4 and the sharp assumption N(σN*)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.
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BibTeX
@misc{geng2026transition,
title = {{BBP Transition for Deformed Inhomogeneous Random Matrices}},
author = {Ruohan Geng and Zhantao He and Dang-Zheng Liu and Guangyi Zou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24796},
eprint = {2610.24796},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24796},
}
RIS
TY - UNPB
TI - BBP Transition for Deformed Inhomogeneous Random Matrices
AU - Ruohan Geng
AU - Zhantao He
AU - Dang-Zheng Liu
AU - Guangyi Zou
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24796
UR - https://arxiv.org/abs/2610.24796
AB - We study finite-rank deformations of inhomogeneous random symmetric/Hermitian matrices H_N=(σ_ijW_ij)_i,j=1^N, whose variance profile satisfies ∑_jσ_ij^2=1. Writing σ_N^*=max_i,jσ_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<α≤ 2 and the sharp assumption σ_N^*(log N)^1/α→ 0, the convergence holds almost surely; (2) under uniformly integrable p-th powers for some p≥ 4 and the sharp assumption N(σ_N^*)^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.
ER -