Let $\alpha, \beta$ be independent d-states (d > 2) Markov chains with the same transition probability matrix $\pi$, and let $P(\alpha_0 =1)=P(\alpha_0 =d)= \frac{1}{2}$, $P(\beta_0 =2)=P(\beta_0 =d−1)= \frac{1}{2}$. Find ALL transition matrices $\pi$ such that $\alpha$, $\beta$ may be successfully coupled.
My attempt
I understand that successful coupling means that, starting at some point $\tau$, the processes are glued ($P(\alpha \neq \beta) = 0, t \geq \tau$). Am I right that here this means chains just have to ever intersect? Then it suffices that there exists N such that the probability of getting into one state is positive after N steps, which can be described as follows: $a_0T (\pi^T)N \pi^N b_0 > 0$. But I can't figure out answer directly in terms of matrix elements, which is required. I understand that $\pi^N > 0$ is enough, but this is too strong condition.