We have a Markov chain on Z with matrix:
$p_{ii+1}=p=1-p_{ii-1}$ for $i\leqslant-1$, $p_{ii-1}=p=1-p_{ii+1}$ for $i\geqslant1$, and $p_{00}=p_{01}=p_{-10}=\frac{1}{3}$.
For which values of p is this chain recurrent?
My attempts/intuition:
I suppose that if $p\geqslant \frac{1}{2}$ then the chain is recurrent but I dont know how to prove it. Maybe it could be helpful to consider $|X_n|$?
I would be thankful for any help.