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

TL;DR

This work analyzes the large-deviation statistics of the largest eigenvalue of deformed Wigner matrices $M=JH+V$ with $H$ drawn from GOE/GUE and a full-rank diagonal perturbation $V$. Using a replica method in the replica-symmetric sector, the authors derive the cumulant generating function $\\phi(s)$ and the large-deviation rate function $\\cal L(\\lambda)$, revealing a delocalized vs localized polymer-like phase transition at a critical coupling $J_c$. They provide explicit forms for $\\phi(s)$, the rate function $\\cal L(\\lambda)$, and the occupation measure $\\rho_\\lambda(v)$, and establish connections to directed polymers and spherical spin glasses; they also extend the framework to quadratic optimization problems in a random field, obtaining corresponding large-deviation results. The results unify and extend previous findings (e.g., McKenna) within a consistent replica-based approach, and offer quantitative predictions for the tail behavior of $\\lambda_{\max}$ and the eigenvector’s overlap with the perturbation, with implications for systems described by deformed random matrices and their applications to spin glasses and polymer models.

Abstract

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.

Figures (1)

• Figure 1: Roots $z_\pm$ of $\lambda=f(z)$ in the three different cases (i) left: delocalized phase $J>J_c$ and delocalized regime (ii) center: delocalized phase $J>J_c$ and localized regime (iii) right: localized phase $J<J_c$. In the delocalized phase $\lambda_{\rm typ}=f(z^*)$ where $z^*$ is the unique minimum of $f(z)$ in the interval $z \in [v_e,+\infty[$. In the localized phase $\lambda_{\rm typ}=\lambda_e=f(v_e)$.