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We study the asymptotic behavior of the spectrum of a random matrix where anon-linearity is applied entry-wise to a Wigner matrix perturbed by a rank-onespike with independent and identically distributed entries. In this setting, weshow that when the signal-to-noise ratio scale as $N^{\frac{1}{2}(1-1/k_\star)}$, where $k_\star$ is the first non-zero generalized informationcoefficient of the function, the non-linear spike model effectively behaves asan equivalent spiked Wigner matrix, where the former spike before thenon-linearity is now raised to a power $k_\star$. This allows us to study thephase transition of the leading eigenvalues, generalizing part of the work ofBaik, Ben Arous and Pech\'e to these non-linear models.

Spectral Phase Transitions in Non-Linear Wigner Spiked Models