I asked this question yesterday but it wasn't well-posed and I made some mistakes in description so I deleted the previous post. Let $G$ be an infinite, connected, undirected graph, and let $\delta_n$ be the set of vertices at distance $n$ from the vertex labelled $0$. With $E_n$ the number of edges joining $\delta\Lambda_n$ to $\delta\Lambda_{n+1}$, show that a random walk on $G$ is recurrent if $∑ \frac{1}{E_n} = \infty$
This question appears in a textbook about Markov Chains on Graphs (Ex. 1.8).
I tried to show that $\sum_{n\in\mathbb{N}} P_0(X_n = 0) \geq \infty$ by somehow transforming these probabilities into something like $P_0(X_{n+1}\in\delta\Lambda_{k+1}|X_{n}\in\delta\Lambda_{k}) \geq \frac{1}{E_k}$