Crested products of Markov chains

TL;DR

This paper introduces two new types of crested products for reversible Markov chains, generalizing existing models and providing a spectral theory, with applications to classical diffusion models and extensions to multiple insects.

Contribution

It defines and analyzes two crested product constructions for Markov chains, extending classical models and developing a comprehensive spectral theory independent of Gelfand pairs.

Findings

Complete spectral theory for crested products

Generalization of classical diffusion models

Spectral decomposition for multi-insect models

Abstract

In this work we define two kinds of crested product for reversible Markov chains, which naturally appear as a generalization of the case of crossed and nested product, as in association schemes theory, even if we do a construction that seems to be more general and simple. Although the crossed and nested product are inspired by the study of Gelfand pairs associated with the direct and the wreath product of two groups, the crested products are a more general construction, independent from the Gelfand pairs theory, for which a complete spectral theory is developed. Moreover, the $k$-step transition probability is given. It is remarkable that these Markov chains describe some classical models (Ehrenfest diffusion model, Bernoulli--Laplace diffusion model, exclusion model) and give some generalization of them. As a particular case of nested product, one gets the classical Insect Markov chain on the ultrametric space. Finally, in the context of the second crested product, we present a generalization of this Markov chain to the case of many insects and give the corresponding spectral decomposition.