Fluctuations of the largest eigenvalues of transformed spiked Wigner matrices

TL;DR

This paper studies the behavior of the largest eigenvalues in a transformed spiked Wigner matrix model, revealing phase transitions and precise distribution limits depending on the signal-to-noise ratio.

Contribution

It establishes BBP-type phase transition results for transformed spiked Wigner matrices, including explicit formulas for eigenvalue fluctuations and distribution limits.

Findings

Largest eigenvalue fluctuations follow Gaussian or Tracy-Widom distributions depending on SNR.

Precise formulas for limiting distributions are derived.

Concentration estimates for eigenvalues are provided in both regimes.

Abstract

We consider a spiked random matrix model obtained by applying a function entrywise to a signal-plus-noise symmetric data matrix. We prove that the largest eigenvalue of this model, which we call a transformed spiked Wigner matrix, exhibits Baik-Ben Arous--P\'ech\'e (BBP) type phase transition. We show that the law of the fluctuation converges to the Gaussian distribution when the effective signal-to-noise ratio (SNR) is above the critical number, and to the GOE Tracy-Widom distribution when the effective SNR is below the critical number. We provide precise formulas for the limiting distributions and also concentration estimates for the largest eigenvalues, both in the supercritical and the subcritical regimes.