Limit Profiles for Separation Distance

TL;DR

This paper investigates the limiting shapes of separation distance profiles in Markov chains, focusing on card shuffles and developing spectral comparison techniques for various random walks.

Contribution

It introduces a spectral comparison method for separation distance and determines limit profiles for key card shuffles and random walks.

Findings

Determined separation distance limit profiles for inverse riffle shuffles and transpositions.

Developed a spectral comparison technique applicable to multiple random walk models.

Demonstrated continuity properties of separation distance profiles.

Abstract

This paper studies limit profiles for the separation distance. A limit profile records the limiting shape of the distance to stationarity inside the cutoff window, at times of the form $t_n+cw_n$. We start with two famous card shuffles, a general setup for inverse riffle shuffles and random transpositions, and we determine their separation distance limit profiles. We then develop a spectral comparison technique and study continuity properties in the style of [Nes24; Nes25], adapted to separation distance. The comparison method is illustrated through random transpositions, as well as random walks on product groups and the hypercube.