Asymptotics of characters of symmetric groups related to Stanley character formula

Abstract

We prove an upper bound for characters of the symmetric groups. In particular, we show that there exists a constant $a>0$ with a property that for every Young diagram $\lambda$ with $n$ boxes, $r(\lambda)$ rows and $c(\lambda)$ columns $$ \left| \frac{\mathrm{Tr}\, \rho^{\lambda}(\pi)}{\mathrm{Tr}\, \rho^{\lambda}(e)} \right| \leq \left[a \max\left(\frac{r(\lambda)}{n},\frac{c(\lambda)}{n},\frac{|\pi|}{n} \right)\right]^{|\pi|}, $$ where $|\pi|$ is the minimal number of factors needed to write $\pi\in S_n$ as a product of transpositions. We also give uniform estimates for the error term in the Vershik-Kerov’s and Biane’s character formulas and give a new formula for free cumulants of the transition measure.