Probability of words summing to $1$ in $S_n$ or $\mathrm{PGL}_2(n)$ $\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\Conj{Conj}$Let $G$ be the symmetric group $S_n$ or the projective general linear group $\PGL_2(n)$.
Let $X$ be a cyclically reduced word in the abstract variables
$x_1, x_2, \ldots,x_k$, i.e. $X$ is a product containing $x_1, x_2, \ldots,x_k$ and their inverses, without any element appearing next to its own inverse in any cyclic permutation. (Only words with length $4$, $6$, $8$ are needed in my research.)
Consider the probability $P$ that the word sums to $1$, with each $x_i$ chosen uniformly and independently from $G$.
Question:
What are the upper bounds of $\log_{|G|}P$?
If $\log_{|G|}P$ converges when $n\to\infty$, what's the value?
Answers are acceptable for either $G=S_n$ or $G=\PGL_2(n)$.
Known:
If there's a variable occurring only once in $X$, then $P$ is exactly $1/|G|$.
If $X=x_1^k$, then the limit is $-1/k$ for symmetric groups by David E Speyer's argument.
As Richard Stanley pointed out, if $X=x_1x_2x_1^{-1}x_2^{-1}$, then $P=|\Conj(G)|/|G|$. ($|\Conj(G)|$ is the number of conjugacy classes of $G$)
The formula $P=|\Conj(G)|/|G|$ holds for the words $x_1x_1x_2x_2$ and $x_1x_2x_1x_2^{-1}$ if all the characters of $G$ are real, and that's exactly the case for $S_n$ and $\PGL_2(n)$.
 A: $\DeclareMathOperator\PGL{PGL}$I believe the best result in this direction is due to M. Larsen & A. Shalev (2012); see this paper. I'll summarize their results here. This doesn't answer the questions whether the limit exists or what its precise value is, though, so this is not a full answer.
Denote the length of the word $X$ by $\ell$ (while $k$ is the size of the alphabet).
As for $S_n$, Proposition 2.3 gives $P < n!^{-\eta}$ for large enough $n$, if $\eta < \frac{2k-1}{4\left((2k)^{2\ell+1}-1\right)}$, so $\log_{|S_n|}P < -\eta. $
As for $\PGL_n(\mathbb{F}_q)$, Proposition 3.3 gives $\log_{|G|}P \le -\eta$ for large enough $n$ (and every $q$), if $\eta < \frac{1}{1800\ell^2}$. (This holds for all classical groups of Lie type).
Shalev and Larsen handle all finite simple groups in their paper. They also refer to this paper, which gives a bound for simple groups $G$ of Lie type of bounded
rank $r$: $ P \le \frac{C}{q}$ where $C = C(r, X)$ is a constant, and $q \ge G^{\epsilon}$ for some $\epsilon = \epsilon(r)$, hence $\log_{|G|}P \le -\eta$ for large enough $G$ if $\eta < \epsilon$.
EDITED: this last paragraph answers the $n=2, q\to\infty$ case.
