TITLE: Proability of a word being fulfilled in the symmetric group
QUESTION [5 upvotes]: Let $S_n$ be the symmetric group on $n$ letters, let $g\in S_n$ be a fixed element of order, say, roughly $n^2$ (e.g. two large disjoint cycles of coprime length), let $w(a,b)$ be a reduced word in letters $a$, $b$, $a^{-1}$, $b^{-1}$.

Question: What is the number of those $h\in S_n$ such that $w(g,h)=1$?

I'm interested in all kinds of information (e.g. asymptotic behaviour when $n\to \infty$, restrictions on $w$, what if order of $g$ is significantly different,...).

REPLY [2 votes]: If you search mathscinet for "word maps" you will be enlightened. See, in particular,
MR2453603 (2010g:20022) 
Larsen, Michael(1-IN); Shalev, Aner(IL-HEBR-E)
Characters of symmetric groups: sharp bounds and applications. (English summary) 
Invent. Math. 174 (2008), no. 3, 645–687. 
20C30 (20E45 20P05 60B15)