Spike Irrelevance and Convergence Rate at the Non-Hermitian BBP Transition
Abstract
We study a rank-one spiked complex Ginibre ensemble A_N = X_N + sigma u u^*, where X_N is an N by N complex Ginibre matrix with independent entries distributed as CN(0,1/N), u is a deterministic unit vector, and sigma is a nonnegative spike strength. We focus on the maximum real part of the eigenvalues and its behavior near the non-Hermitian BBP transition. For every fixed subcritical spike strength sigma < 1, we show that the maximum real part has the same Gumbel limit as in the unspiked ensemble, after the same centering and scaling. We also derive an explicit convergence-rate correction of order 1/log N for the unspiked Gumbel law. At the critical point sigma = 1, we establish an upper bound for the rightmost eigenvalue at the N^{-1/4} scale using isotropic resolvent estimates and a net argument.
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BibTeX
@misc{wu2026spike,
title = {{Spike Irrelevance and Convergence Rate at the Non-Hermitian BBP Transition}},
author = {Yuehan Wu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19221},
eprint = {2609.19221},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19221},
}
RIS
TY - UNPB
TI - Spike Irrelevance and Convergence Rate at the Non-Hermitian BBP Transition
AU - Yuehan Wu
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.19221
UR - https://arxiv.org/abs/2609.19221
AB - We study a rank-one spiked complex Ginibre ensemble A_N = X_N + sigma u u^*, where X_N is an N by N complex Ginibre matrix with independent entries distributed as CN(0,1/N), u is a deterministic unit vector, and sigma is a nonnegative spike strength. We focus on the maximum real part of the eigenvalues and its behavior near the non-Hermitian BBP transition. For every fixed subcritical spike strength sigma < 1, we show that the maximum real part has the same Gumbel limit as in the unspiked ensemble, after the same centering and scaling. We also derive an explicit convergence-rate correction of order 1/log N for the unspiked Gumbel law. At the critical point sigma = 1, we establish an upper bound for the rightmost eigenvalue at the N^{-1/4} scale using isotropic resolvent estimates and a net argument.
ER -