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

We study the probability distribution function $P(\lambda)$ of the largest eigenvalue $\lambda_{\rm max}$ 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 ${\cal L}(\lambda)$ which describes the upper large deviation tail $P(\lambda) \sim e^{- \beta N {\cal L}(\lambda) }$. We also obtain the moment generating function $\langle e^{N s {\lambda}_{\max} } \rangle \sim e^{N \phi(s)}$ 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.