The Limit Profile of Star Transpositions

TL;DR

This paper characterizes the asymptotic behavior of the total variation distance for star transpositions at a specific time scale, revealing a Poisson-based limit profile and establishing a comparison technique for reversible Markov chains.

Contribution

It introduces a new method for comparing the limit profiles of reversible Markov chains sharing the same stationary distribution and eigenbasis, and applies it to star transpositions.

Findings

Limit profile of star transpositions converges to a Poisson distribution difference.

Star transpositions and random transpositions share the same limit profile at cutoff times.

Develops a technique for comparing reversible Markov chains' limit profiles.

Abstract

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.