This is mainly a request for a reference. I am looking for a book or a paper wherein it is proved that if $G$ is a finite, simple, connected graph, then the random walk process is recurrent. I have seen this mentioned in the literature and have found a few proofs from electrical engineering where it is proved for large networks but I am looking for the simpler case of finite connected graphs, preferably from a graph or probability theory corner. Any help would be appreciated.
(I feel like this is probably a repost ... did my due dilligence but could not find a previous listing of this question)
If $|G|=n>1,$ then then for each nodes $a,b$ the probability $p(a,b)$ that, starting f from $a,$ you get to $b$ within any of the next $n$ steps is a positive value. We have that $p(a,b)>0$ for all pairs $a,b\in G$ since $G$ is connected.
Then, for any fixed $b$, we'd take $q(b)=\min_a p(a,b)>0.$ So, at any time, you will get to $b$ in the next $n$ steps with at least probability $q(b)>0.$ Now, over an infinite time, you have probability $0$ of reaching $b$ only finitely many time.