I know that if a Markov chain shows rotational symmetry, the stationary measure must respect the rotationary symmetry. So the obvious candidate for stationary measure is the uniform measure, where we assign the mass of 1 to each state. For example, a pawn moving counter-clockwise on a circle with 40 positions. At each time step, we move the pawn by as many positions as the sum of the face values of two fair dice. So, we can choose (1, 1, ... , 1) as our stationary measure. For the Markov Chain in the question, the translation probabilities do not change if another vertex is chosen to be the origin. This suggest that the stationary measure also does not change under spatial shift of coordinates. I suspect that the construction of the stationary measure should include the probability measure μ. Can someone help me with how I derive the stationary measure?