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 IN that lie in an arbitrary interval of the real line for N×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≫1), leading to the correct logarithmic scaling of the variance 〈IN2〉−〈IN〉2=O(ln⁡N) as well as to an analytical expression for the O(1/N) correction to the average 〈IN〉/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 IN in non-invariant random matrix ensembles, where the joint distribution of eigenvalues is not known.