Large deviations of the largest eigenvalue for deformed GOE/GUE random matrices via replica
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908588433211392 |
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| author | Doussal, Pierre Le |
| author_facet | Doussal, Pierre Le |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_12148 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Large deviations of the largest eigenvalue for deformed GOE/GUE random matrices via replica Doussal, Pierre Le Statistical Mechanics Mathematical Physics Probability 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. |
| title | Large deviations of the largest eigenvalue for deformed GOE/GUE random matrices via replica |
| topic | Statistical Mechanics Mathematical Physics Probability |
| url | https://arxiv.org/abs/2503.12148 |