Asymptotics of characters of symmetric groups related to Stanley character formula

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We 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 λwith n boxes, r(λ) rows and c(λ) columns |Tr ρλ(π) / Tr ρλ(e)| < [a max(r(λ)/n, c(λ)/n,|π|/n)]|π|, where |π| is the minimal number of factors needed to write π∈ Sₙ 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.