The Diagonal Approximation for Holographic R\'{e}nyi Entropies

Recently Dong, Rath and Kudler-Flam proposed a modified cosmic brane prescription for computing the R\'{e}nyi entropy $S_\alpha$ of a holographic system in the presence of multiple extremal surfaces. This prescription was found by assuming a diagonal approximation, where the R\'{e}nyi entropy is computed after first measuring the areas of all extremal surfaces. We derive this diagonal approximation and show that it accurately computes R\'{e}nyi entropies up to $O(\log G)$ corrections. For $\alpha<1$, this allows us to derive the modified cosmic brane prescription, which differs from the original cosmic brane prescription at leading order in $G$. For $\alpha>1$, it leads to the original cosmic brane prescription without needing to assume that replica symmetry is unbroken in the bulk.