I am trying to show that; If $X_n, ~ n = 0,1,...$ is a simple random walk on d-regular graph (finite or infinite) starts at $X_0$, after any even number of steps the most likely position is $X_0$.
I can see that since it is regular then the stationary distribution is $~Uniform~$ and I could prove the convergence to uniform distribution, but I don't see why the origin would have higher probability when we have uniform distribution on all vertices.
Can you help me showing that?
Thank you in advance.