Tridiagonalized GUE matrices are a matrix model for labeled mobiles

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.Comment: 42 pages, LaTeX, 17 figures. The present paper completely supercedes arXiv1203.3185 in terms of methods but addresses a different proble

Character and class parameters from entries of character tables of symmetric groups

If all of the entries of a large $S_n$ character table are covered up and you are allowed to uncover one entry at a time, then how can you quickly identify all of the indexing characters and conjugacy classes? We present a fast algorithmic solution that works even when n is so large that almost none of the entries of the character table can be computed. The fraction of the character table that needs to be uncovered has exponential decay, and for many of these entries we are only interested in whether the entry is zero

Covering numbers for characters of symmetric groups

If $n>4$ and $c(\theta)$ denotes the set of irreducible constituents of a character $\theta$, then $c(\chi^k)={\rm Irr}(S_n)$ for all nonlinear $\chi\in {\rm Irr}(S_n)$ if and only if $k\geq n-1$