Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 4 / 26J / b / Solution

The second continuous-time Markov chain runs in the reverse direction at rate $\rho$, as in the right diagram. Its nonzero eigenvalues have real part $-3\rho /2$, and its transition probabilities tend to $1/3$ for each vertex. Independence therefore givesThus the limit exists for every pair of starting vertices and equals $\boxed{1/3}$.

One can also see the limit directly without spectral multiplication. For fixed starting vertices, $X_t-Y_t$ modulo three advances with each jump of either flea. Its jump count is the sum of independent Poisson processes with rates $1$ and $\rho$, hence is Poisson with rate $1+\rho$ (multiply their generating functions). The formula from part (a), with $t$ replaced by $(1+\rho )t$ and the appropriate initial residue, gives $p(t)$ and the same limit.