## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 19, Number 1 (2009), 414-453.

### Crested products of Markov chains

Daniele D’Angeli and Alfredo Donno

#### 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.

#### Article information

**Source**

Ann. Appl. Probab., Volume 19, Number 1 (2009), 414-453.

**Dates**

First available in Project Euclid: 20 February 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1235140344

**Digital Object Identifier**

doi:10.1214/08-AAP546

**Mathematical Reviews number (MathSciNet)**

MR2498683

**Zentralblatt MATH identifier**

1166.60321

**Subjects**

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 15A69: Multilinear algebra, tensor products 05E30: Association schemes, strongly regular graphs 05C25: Graphs and abstract algebra (groups, rings, fields, etc.) [See also 20F65] 43A85: Analysis on homogeneous spaces

**Keywords**

Reversible Markov chain crested product nested product crossed product spectral theory association schemes Gelfand pairs

#### Citation

D’Angeli, Daniele; Donno, Alfredo. Crested products of Markov chains. Ann. Appl. Probab. 19 (2009), no. 1, 414--453. doi:10.1214/08-AAP546. https://projecteuclid.org/euclid.aoap/1235140344