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 $O(1/N)$ ($N \gg 1$), leading to the correct logarithmic scaling of the variance $\langle I_{N}^{2} \rangle - \langle I_N \rangle^2 = O(\ln N)$ as well as to an analytical expression for the $O(1/N)$ correction to the average $\langle I_N \rangle/N$. 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.