Let $\Gamma$ be a simple, locally finite, acyclic graph. Let $v\_0$ be some vertex in $\Gamma$.


We let $X\_n$ denote the simple random walk on $\Gamma$ where $X\_0 = v\_0$. If we almost surely have $\limsup\_{n\to\infty} \frac{d(X\_0,X\_n)}{n}=0$ (using the path metric), does it follow that almost surely $X\_t = X\_0$ for some $t> 0$?



Does there exist a non-recurrent acyclic graph with sublinear expansion?
For a spherically symmetric tree (where all vertices at the same distance from the root have the same degree) it is well known that transience is equivalent to $ \sum\_n |T\_n|^{-1} <\infty$, (where $T\_n$ is level $n$ of the tree) since that means the resistance from the root to infinity is finite. On the other hand, the Varopolous-Carne inequality (see e.g. [3]) implies the speed is zero if the tree has polynomial growth.
So if every vertex at a level which is a power of 2 has 4 children and all other vertices have a single child, then the tree is transient but the speed of SRW is zero.


[1] Lyons, Russell. "Random walks and percolation on trees." The annals of Probability (1990): 931-958.


[2] Benjamini, Itai, and Yuval Peres. "Random walks on a tree and capacity in the interval." In Annales de l'IHP Probabilités et statistiques, vol. 28, no. 4, pp. 557-592. 1992.


[3] Lyons, R., & Peres, Y. (2017). Probability on Trees and Networks (Cambridge Series in Statistical and Probabilistic Mathematics). Cambridge: Cambridge University Press. doi:10.1017/9781316672815 <https://rdlyons.pages.iu.edu/prbtree/book_online.pdf>


