Past exam of the probability course of the University of Oxford / 2021 / 3 / b / i / Solution

Fix any two integers $i,j$. By Bézout's identity, there are integers $u,v$ with $au-bv=j-i$. Replacing $(u,v)$ by $(u+kb,v+ka)$ leaves this identity unchanged. Choose $k$ large enough that both entries are nonnegative. A path with $u+kb$ upward steps and $v+ka$ downward steps then takes $i$ to $j$, with strictly positive probability $p^{u+kb}{q}^{v+ka}$. There are no boundary restrictions on this state space.

This applies in both directions, so every pair of states communicates. Hence