A Recursion Formula for the Moments of the Gaussian Orthogonal Ensemble

– We present an analogue of the Harer-Zagier recursion formula for the moments of the Gaussian Orthogonal Ensemble in the form of a five term recurrence equation. The proof is based on simple Gaussian integration by parts and differential equations on Laplace transforms. A similar recursion formula holds for the Gaussian Symplectic Ensemble. As in the complex case, the result is interpreted as a recursion formula for the number of 1-vertex maps in locally orientable surfaces with a given number of edges and faces. The formula is also applied to a sharp bound on the largest eigenvalue of the Gaussian Orthogonal Ensemble and, by moment comparison, of families of Wigner matrices.