Asymptotics of characters of symmetric groups related to Stanley character formula

20 pagesInternational audienceWe prove an upper bound for characters of the symmetric groups. Namely, 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 |\chi^\lambda(\pi)| < [a max(r(\lambda)/n, c(\lambda)/n,|\pi|/n) ]^{|\pi|}, where |\pi| is the minimal number of factors needed to write \pi\in S_n as a product of transpositions and \chi^\lambda(\pi)= Tr \rho^\lambda(\pi) / Tr \rho^\lambda(e) is the character of the symmetric group. 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