Suppose $G$ is a finitely-generated non-elementary hyperbolic group and consider a symmetric random walk on the Cayley graph $\text{Cay}(G,S)$ with generating set $S$. Denote the set of points $B_{\ell} = \{ g: d(e,g) = \ell\}$ where $d$ is the word metric.
Let $X_n$ be the random variable denoting the position of an $n$-step symmetric random walk which starts at $e$ on the Cayley graph of a non-amenable group, I know that it has finite drift $\frac{1}{n}\lim_{n \to \infty} \mathbb{E} d(X_n,e) = v > 0$, and from (https://arxiv.org/pdf/2101.08222) $d(X_n,e)$ appears to concentrate about $v n$ if $G$ is hyperbolic.
I am wondering about properties of $\mathbb{P}(X_n = g)$ if $g \in B_{v n}$. In particular, I would not expect $\mathbb{P}(X_n = g)$ to be uniform over all points $g \in B_{v n}$ (since roughly this would require $\text{Cay}(G,S)$ to have a great number of isometries). For example, if $G$ is a free group then $\mathbb{P}(X_n = g)$ is uniform over all $g \in B_{v n}$.
Are there results in the literature where I could look to quantify how far from uniform this distribution can be if $G$ is hyperbolic? Or, are there any simple examples of $G$ where this distribution is very "far" from uniform?
