The Limit Profile of Star Transpositions

We prove that the limit profile of star transpositions at time $t= n \log n +cn$ is equal to $d_{\text{T.V.}}(\text{Poiss}(1+e^{-c}), \text{Poiss}(1))$. We prove this by developing a technique for comparing the limit profile behavior of two reversible Markov chains on the same space, that share the same stationary distribution and eigenbasis. We then compare the limit profile of star transpositions to the limit profile of random transpositions, as studied in \cite{Teyssier}, and prove that they have the same limit profile at the respective cutoff times.