Weak lumping of left-invariant random walks on left cosets of finite groups

TL;DR

This paper characterizes when left-invariant random walks on finite groups weakly lump on cosets, providing conditions on initial distributions and weights, with applications to card shuffling Markov chains.

Contribution

It offers a complete characterization of weak lumping conditions for left-invariant walks on finite groups, including explicit linear equations for abelian subgroups.

Findings

Characterization of initial distributions and weights for weak lumping

Explicit transition matrices for the induced Markov chain

Application to specific card shuffling processes

Abstract

Let $G$ be a finite group and let $H$ be a subgroup of $G$. The left-invariant random walk driven by a probability measure $w$ on $G$ is the Markov chain in which from any state $x \in G$, the probability of stepping to $xg \in G$ is $w(g)$. The initial state is chosen randomly according to a given distribution. The walk is said to lump weakly on left cosets if the induced process on $G/H$ is a time-homogeneous Markov chain. We characterise all the initial distributions and weights $w$ such that the walk is irreducible and lumps weakly on left cosets, and determine all the possible transition matrices of the induced Markov chain. In the case where $H$ is abelian we refine our main results to give a necessary and sufficient condition for weak lumping by an explicit system of linear equations on $w$, organized by the double cosets $HxH$. As an application we consider shuffles of a deck of $n$ cards such that repeated observations of the top card form a Markov chain. Such shuffles include the random-to-top shuffle, and also, when the deck is started in a uniform random order, the top-to-random shuffle. We give a further family of examples in which our full theory of weak lumping is needed to verify that the top card sequence is Markov.