Weak lumping of left-invariant random walks on left cosets of finite groups
| Autor: | Crane, Edward, Gutiérrez, Álvaro, Russell, Erin, Wildon, Mark |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 85 pages, 3 figures |
| Databáze: | arXiv |
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