Tridiagonalized GUE matrices are a matrix model for labeled mobiles

TL;DR

This paper introduces a novel approach to calculating the leading coefficient of GUE matrix cumulants by tridiagonalization, cluster expansion, and group-theoretic interpretation, linking random matrices to labeled mobiles.

Contribution

It presents a new method combining matrix tridiagonalization and statistical mechanics techniques to connect GUE matrices with labeled mobiles, offering an alternative to traditional combinatorial methods.

Findings

Established a link between GUE matrix cumulants and labeled mobiles.

Developed a tridiagonalization and cluster expansion framework for analysis.

Reconciled group-theoretic structures with combinatorial models.

Abstract

It is well-known that the number of planar maps with prescribed vertex degree distribution and suitable labeling can be represented as the leading coefficient of the $\frac{1}{N}$-expansion of a joint cumulant of traces of powers of an $N$-by-$N$ GUE matrix. Here we undertake the calculation of this leading coefficient in a different way. Firstly, we tridiagonalize the GUE matrix in the manner of Trotter and Dumitriu-Edelman and then alter it by conjugation to make the subdiagonal identically equal to $1$. Secondly, we apply the cluster expansion technique (specifically, the Brydges-Kennedy-Abdesselam-Rivasseau formula) from rigorous statistical mechanics. Thirdly, by sorting through the terms of the expansion thus generated we arrive at an alternate interpretation for the leading coefficient related to factorizations of the long cycle $(12\cdots n)\in S_n$. Finally, we reconcile the group-theoretical objects emerging from our calculation with the labeled mobiles of Bouttier-Di Francesco-Guitter.