For $g$ an element of a group and $\chi$ an irreducible character, there are two easy bounds for the character value $\chi(g)$: First, the bound $|\chi(g)|\leq \chi(1)$ by the dimension of the representation and, second, the bound $|\chi(g) |\ \leq \sqrt{ |Z(g)|}$ by the centralizer arising from the formula $\sum_{\chi} |\chi(g)|^2= |Z(g)|$.
I am interested on upper bounds for characters of the symmetric group that improve slightly on the second bound.
More specifically, fix $\delta> 0$ small. I want to know for which $g\in S_n$ there exists an irreducible character $\chi$ with $|\chi(g)| > \sqrt{ |Z(g)|} e^{- \delta \sqrt{n}}$.
Recall that for a permutation $g\in S_n$ with $m_k$ cycles of size $k$ for all $k$, so that $\sum_k m_k k =n$, we have $|Z(g)| = \prod_k (k^{m_k} m_k!)$.
For example, known upper bounds for the dimension of representations of the symmetric group imply there does not exist such a character for $g$ the identity. On the other hand, such a character does exist if $g$ is an $n$-cycle, since the right side is less than $1$. I suspect and hope that this can only happen for $g$ containing some relatively large cycles.
I found works in the literature giving bounds for $\chi(g)$ that improve on the $|\chi(g)|\leq \chi(1)$ by a multiplicative factor (Asymptotics of characters of symmetric groups related to Stanley character formula by Féray and Śniady) or a power (Characters of symmetric groups: sharp bounds and applications by Michael Larsen and Aner Shalev) but it's not obvious if it's possible to transform them into a bound of the form I need.
