Exact moments of the Sachdev-Ye-Kitaev model up to order $1/N^2$.

We analytically evaluate the moments of the spectral density of the $q$-body Sachdev-Ye-Kitaev (SYK) model, and obtain order $1/N^2$ corrections for all moments, where $N$ is the total number of Majorana fermions. To order $1/N$, moments are given by those of the weight function of the Q-Hermite polynomials. Representing Wick contractions by rooted chord diagrams, we show that the $1/N^2$ correction for each chord diagram is proportional to the number of triangular loops of the corresponding intersection graph, with an extra grading factor when $q$ is odd. Therefore the problem of finding $1/N^2$ corrections is mapped to a triangle counting problem. Since the total number of triangles is a purely graph-theoretic property, we can compute them for the $q=1$ and $q=2$ SYK models, where the exact moments can be obtained analytically using other methods, and therefore we have solved the moment problem for any $q$ to $1/N^2$ accuracy. The moments are then used to obtain the spectral density of the SYK model to order $1/N^2$. We also obtain an exact analytical result for all contraction diagrams contributing to the moments, which can be evaluated up to eighth order. This shows that the Q-Hermite approximation is accurate even for small values of $N$.