$\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\Conj{Conj}$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 $\PGL\_2(n)$.



Probability of words summing to $1$ in $S_n$ or $\mathrm{PGL}_2(n)$
$\DeclareMathOperator\PGL{PGL}$I believe the best result in this direction is due to M. Larsen & A. Shalev (2012); see [this](https://www.researchgate.net/publication/256735756_Fibers_of_word_maps_and_some_applications) 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](https://people.math.carleton.ca/%7Ejdixon/Residual.pdf) 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.


