Replica-symmetric approach to the typical eigenvalue fluctuations of Gaussian random matrices

Abstract

We discuss an approach to compute the first and second moments of the number of eigenvalues $I_N$ that lie in an arbitrary interval of the real line for $N\times N$ Gaussian random matrices. The method combines the standard replica-symmetric theory with a perturbative expansion of the saddle-point action up to $O(1/N)$ ($N\gg 1$), leading to the correct logarithmic scaling of the variance $⟨I_N^2⟩-⟨I_N{⟩}^2=O(\ln N)$ as well as to an analytical expression for the $O(1/N)$ correction to the average $⟨I_N⟩/N$. Standard results for the number variance at the local scaling regime are recovered in the limit of a vanishing interval. The limitations of the replica-symmetric method are unveiled by comparing our results with those derived through exact methods. The present work represents an important step to study the fluctuations of $I_N$ in non-invariant random matrix ensembles, where the joint distribution of eigenvalues is not known.

17 pages, 2 figures

References in corpus (6)

• Phase transitions and edge scaling of number variance in Gaussian random matrices
• Number statistics for $\beta$-ensembles of random matrices: applications to trapped fermions at zero temperature
• Level compressibility for the Anderson model on regular random graphs and the eigenvalue statistics in the extended phase
• Large deviation function for the number of eigenvalues of sparse random graphs inside an interval
• Spectral order statistics of Gaussian random matrices: large deviations for trapped fermions and associated phase transitions
• Index statistical properties of sparse random graphs

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