Large deviations of the largest eigenvalue for deformed GOE/GUE random matrices via replica

We study the probability distribution function $P(λ)$ of the largest eigenvalue $λ_{\rm max}$ of $N \times N$ random matrices of the form $H + V$, where $H$ belongs to the GOE/GUE ensemble and $V$ is a full rank deterministic diagonal perturbation. This model is related to spherical spin glasses and semi-discrete directed polymers. In the large $N$ limit, using the replica method introduced in Ref. \cite{TrivializationUs2014}, we obtain the rate function ${\cal L}(λ)$ which describes the upper large deviation tail $P(λ) \sim e^{- βN {\cal L}(λ) }$. We also obtain the moment generating function $\langle e^{N s λ_{\max} } \rangle \sim e^{N ϕ(s)}$ and the overlap of the optimal eigenvector with the perturbation $V$. For suitable $V$, a transition generically occurs in the rate functions. For the GUE it has a direct interpretation as a localisation transition for tilted directed polymers with competing columnar and point disorder. Although in a different form, our results are consistent with those obtained recently by Mc Kenna in \cite{McKenna2021}. Finally, we consider briefly the quadratic optimisation problem in presence of an additional random field and obtain its large deviation rate function, although only within the replica symmetric phase.