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We discuss an approach to compute the first and second moments of the numberof eigenvalues $I_N$ that lie in an arbitrary interval of the real line for $N\times N$ Gaussian random matrices. The method combines the standardreplica-symmetric theory with a perturbative expansion of the saddle-pointaction up to $O(1/N)$ ($N \gg 1$), leading to the correct logarithmic scalingof 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 theaverage $\langle I_N \rangle/N$. Standard results for the number variance atthe local scaling regime are recovered in the limit of a vanishing interval.The limitations of the replica-symmetric method are unveiled by comparing ourresults with those derived through exact methods. The present work representsan important step to study the fluctuations of $I_N$ in non-invariant randommatrix ensembles, where the joint distribution of eigenvalues is not known.

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