Question: Is there a sequence $(\delta_n)_n$ of real numbers with $\delta_n \to 0$ as $n \to \infty$, such that the following holds:
Let $F$ be a free group on two generators, let $F \curvearrowright X$ be a transitive action on an infinite set, and let $x \in X$. Then, the probability that a random word of length $n$ in $F$ fixes $x \in X$ is smaller than $\delta_n$.
I would like to consider random unreduced words (so that there are $4^n$ such words of length $n$ and each is equally likely), but probably this does not matter much. It corresponds to the nearest neighbor random walk on the Schreier graph corresponding to the action of $F$ on $X$.
It is clear that for each individual action $x \in X$, the return probability decays; and it seems plausible that this happens uniformly over all actions.