Generalized crested products of Markov chains

TL;DR

This paper introduces a generalized crested product of Markov chains, providing spectral analysis, transition probabilities, and connections to classical models and group theory, broadening the understanding of Markov chain structures.

Contribution

It defines a new generalized Markov chain model, analyzes its spectral properties, and links it to classical diffusion models and group representations.

Findings

Spectral decomposition of the generalized crested product is developed.

Transition probabilities for the k-step process are explicitly derived.

Connections to classical models like Ehrenfest and Insect Markov are established.

Abstract

We define a finite Markov chain, called generalized crested product, which naturally appears as a generalization of the first crested product of Markov chains. A complete spectral analysis is developed and the $k$-step transition probability is given. It is important to remark that this Markov chain describes a more general version of the classical Ehrenfest diffusion model. As a particular case, one gets a generalization of the classical Insect Markov chain defined on the ultrametric space. Finally, an interpretation in terms of representation group theory is given, by showing the correspondence between the spectral decomposition of the generalized crested product and the Gelfand pairs associated with the generalized wreath product of permutation groups.