Word Measures on Symmetric Groups

Fix a word $w$ in free group $F$ on $r$ generators. A $w$-random permutation the symmetric $S_N$ is obtained by sampling independent uniformly random permutations $\sigma_{1},\ldots,\sigma_{r}\in S_{N}$ and evaluating $w\left(\sigma_{1},\ldots,\sigma_{r}\right)$. In [arXiv:1104.3991, arXiv:1202.3269] it was shown that average number of fixed points $1+\theta\left(N^{1-\pi\left(w\right)}\right)$, where $\pi\left(w\right)$ smallest rank subgroup $H\le F$ containing as non-primitive element. We show plays role estimates all stable characters groups. particular, we for $t\ge2$, $t$-cycles $\frac{1}{t}+O\left(N^{-\pi\left(w\right)}\right)$. As an application, prove every $s$, $\varepsilon>0$ large enough $r$, Schreier graphs with generators depicting action $S_{N}$ $s$-tuples, have second eigenvalue at most $2\sqrt{2r-1}+\varepsilon$ asymptotically almost surely. An important ingredient this work systematic study not-necessarily connected Stallings core graphs.