Replica theory for the rate functional of the empirical spectral distribution function of diluted Hermitian matrices (opens in new tab)

We develop a replica-based framework for the scaled cumulant-generating functional of the empirical spectral distribution function $i_C$ of diluted Hermitian random matrices. Within a replica-symmetric saddle-point assumption, this construction yields a candidate rate functional for fluctuations of $i_C$. As an illustrative application, we consider adjacency matrices of unweighted Erd\H{o}s-R\'enyi random graphs with mean degree $c$. We derive...