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 $1+\theta\left(N^{1-\pi\left(w\right)}\right)$...