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

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 $O(1\mathrm{/}N)$ ($N\gg 1$), 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 $O(1\mathrm{/}N)$ correction to the average $⟨I_N⟩\mathrm{/}N$. 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.