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It is well-known that the number of planar maps with prescribed vertex degreedistribution and suitable labeling can be represented as the leadingcoefficient of the $\frac{1}{N}$-expansion of a joint cumulant of traces ofpowers of an $N$-by-$N$ GUE matrix. Here we undertake the calculation of thisleading coefficient in a different way. Firstly, we tridiagonalize the GUEmatrix in the manner of Trotter and Dumitriu-Edelman and then alter it byconjugation to make the subdiagonal identically equal to $1$. Secondly, weapply the cluster expansion technique (specifically, theBrydges-Kennedy-Abdesselam-Rivasseau formula) from rigorous statisticalmechanics. Thirdly, by sorting through the terms of the expansion thusgenerated we arrive at an alternate interpretation for the leading coefficientrelated to factorizations of the long cycle $(12\cdots n)\in S_n$. Finally, wereconcile the group-theoretical objects emerging from our calculation with thelabeled mobiles of Bouttier-Di Francesco-Guitter.

Tridiagonalized GUE matrices are a matrix model for labeled mobiles