A Tridiagonal Approach to Matrix Integrals

Physicists in the 70’s starting with ’t Hooft established that the number of suitably labeled planar maps with prescribed vertex degree distribution can be represented as the leading coefficient of the 1 N -expansion of a joint cumulant of traces of powers of a standard N-by-N GUE matrix. Here we undertake the calculation of this leading coefficient in a different way, namely, after first tridiagonalizing the GUE matrix à la Trotter and DumitriuEdelman and then “de-symmetrizing,” we apply the cluster expansion technique (specifically, the Brydges-Kennedy-Abdesselam-Rivasseau formula) from rigorous statistical mechanics. We thus arrive at our main result, which is an alternate combinatorial interpretation for the leading coefficient in terms of edge-labeled planar trees equipped with a vertex-four-coloring subject to certain simple rules. Objects of the latter type, without matching up exactly, bear a family resemblance to the well-labeled trees already in common use to enumerate planar maps, e.g., those of Cori-Vauquelin and of Schaeffer. By using a combinatorial insight of Goulden-Jackson, we can straightforwardly reconcile our main result with a formula of Tutte from the 60’s counting rooted planar maps with a prescribed Eulerian (all degrees even) vertex degree distribution. (But our main result has no Eulerian hypotheses.) Ultimately the contribution of the paper is simply to demonstrate the close connection of tridiagonalized GUE matrices to well-labeled trees.