Spike Irrelevance and Convergence Rate at the Non-Hermitian BBP Transition

Yuehan Wu

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arXiv · 2609.19221

Spike Irrelevance and Convergence Rate at the Non-Hermitian BBP Transition

Yuehan Wu

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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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Yuehan Wu. 2026-09-16. Spike Irrelevance and Convergence Rate at the Non-Hermitian BBP Transition. https://arxiv.org/abs/2609.19221

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