标题： 去变形 GOE/GUE 随机矩阵的最大特征值的大偏差通过复制 显示英文标题

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

摘要： 我们研究了形如$H + V$的$N \times N$随机矩阵的最大特征值$\lambda_{\rm max}$的概率分布函数$P(\lambda)$，其中$H$属于 GOE/GUE 系列，$V$是一个满秩的确定性对角扰动。 该模型与球面自旋玻璃和半离散定向聚合物有关。 在大$N$极限下，使用参考文献中引入的复本方法。 \cite{TrivializationUs2014}，我们得到速率函数${\cal L}(\lambda)$，它描述了上部大偏差尾部$P(\lambda) \sim e^{- \beta N {\cal L}(\lambda) }$。 我们还得到了矩生成函数$\langle e^{N s {\lambda}_{\max} } \rangle \sim e^{N \phi(s)}$和最优特征向量与扰动的重叠$V$。 对于合适的$V$，速率函数中通常会发生相变。 对于GUE，它可以直接解释为具有竞争性柱状和点无序的倾斜定向聚合物的局域化相变。 尽管形式不同，我们的结果与Mc Kenna最近在\cite{McKenna2021}中得到的结果一致。 最后，我们简要地考虑了存在额外随机场时的二次优化问题，并在其大偏差速率函数方面得到了结果，尽管仅限于副本对称相内。 显示英文摘要

摘要： 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.