Comparing limit profiles of reversible Markov chains

We introduce a technique for comparing the limit profile behavior of two reversible, commuting Markov chains on the same space, that share the same stationary distribution. We apply this technique to prove that the limit profile of star transpositions at time t = n log n + cn is equal to d_T.V. (Poiss(1 + e^−c), Poiss(1)) by comparing to the limit profile of random transpositions, as studied in [29]. We also provide examples of important commuting Markov chains, whose limit profile behavior is unknown, which could give new directions for research.