Finite-size corrections to the spectrum of regular random graphs: An analytical solution

We develop a thorough analytical study of the O(1/N) correction to the spectrum of regular random graphs with N->? nodes. The finite-size fluctuations of the resolvent are given in terms of a weighted series over the contributions coming from loops of all possible lengths, from which we obtain the isolated eigenvalue as well as an analytical expression for the O(1/N) correction to the continuous part of the spectrum. The comparison between this analytical formula and direct diagonalization results exhibits an excellent agreement, confirming the correctness of our expression.