proof:**]{} It has just been discovered that there is an error in \[b\].AL:uni/. Namely, there is a gap in the proof that all unimodular random rooted graphs are random weak limits of finite graphs. This makes the proofs of certain results here incomplete. \[r\].explicit/, \[p\].positive/, and \[t\].zero/ will be complete when restricted to random weak limits of finite graphs, rather than claimed for all unimodular random rooted graphs.

We denote a (multi-)graph $\gh$ with vertex set $\verts$ and edge set $\edges$ by $\gh = (\verts, \edges)$. When there is more than one graph under discussion, we write $\verts(\gh)$ or $\edges(\gh)$ to avoid ambiguity. We denote the degree of a vertex $x$ in a graph $\gh$ by $\deg_\gh(x)$. Unless stated otherwise, we assume all degrees finite. [**Simple random walk**]{} on $\gh$ is the Markov chain whose state space is $\verts$ and whose transition probability from $x$ to $y$ equals the number of edges joining $x$ to $y$ divided by $\deg_\gh(x)$.

An infinite path in a tree that starts at any vertex and does not backtrack is called a [**ray**]{}. Two rays are [**equivalent**]{} if they have infinitely many vertices in common. An equivalence class of rays is called an [**end**]{}. More 