In this chapter we explore a theory which gives an alternative approach to some of the diffusion processes presented in the introduction (namely the random walk on the discrete circle, the Ehrenfest and the Bernoulli––Laplace models). In some sense, this can be regarded as a theory of (finite) Gelfand pairs without group theory. Thus, Sections 5.1, 5.2, and 5.3 (as well as Sections 6.1 and 6.3 in the next chapter) do not rely on group representation theory and can be read independently of Chapters 3 and 4. The connection with group theory will be presented in the final part of Section 5.4 and in Section 6.2.

Harmonic analysis on distance-regular graphs

In this section we focus our attention on a remarkable class of finite graphs for which it is possible to develop a nice harmonic analysis. Our exposition is inspired to the monographs by Bailey and by Bannai and Ito. We would like to mention that during our preparation of this book we attended a minicourse by Rosemary Bailey on association schemes which undoubtedly turned out to be very useful and stimulating for us.

We shall denote by X a finite, connected (undirected) graph without self-loops. Recall that given two vertices x, y ∈ X, their distance d(x, y) is the length of the shortest path joining x and y. This way, (X, d) becomes a metric space.

In September 2003 we started writing a research expository paper on “Finite Gelfand pairs and their applications to probability and statistics” for the proceedings of a conference held in Batumi (Georgia). After a preliminary version of that paper had been circulated, we received several emails of appreciation and encouragement from experts in the field. In particular, Persi Diaconis suggested that we expand that paper to a monograph on Gelfand pairs. In his famous 1988 monograph “Group representations in probability and statistics” there is a short treatement of the theory of Gelfand pairs but, to his and our knowledge, no book entirely dedicated to Gelfand pairs was ever written. We thus started to expand the paper, including some background material to make the book self-contained, and adding some topics closely related to the kernel of the monograph. As the “close relation” is in some sense inductive, we pushed our treatement much further than what Persi was probably expecting. In all cases, we believe that our monograph is in some sense unique as it assembles, for the first time, the various topics that appear in it.

The book that came out is a course in finite harmonic analysis. It is completely self-contained (it only requires very basic rudiments of group theory and of linear algebra). There is also a large number of exercises (with solutions or generous hints) which constitute complements and/or further developments of the topics treated.

This chapter contains an exposition on the theory of (finite) Gelfand pairs and their spherical functions. This theory originally was developed in the setting of Lie groups with the seminal paper by I. M. Gelfand (see also) another earlier contribution is the paper by Godement.

Expositions of the theory in the setting of locally compact and/or Lie groups are in Dieudonnè's treatise on analysis and in the monographs by: Dym and McKean, Faraut, Figà-Talamanca and Nebbia, Helgason, Klimyk and Vilenkin, Lang and Ricci. See also the papers by Bougerol.

Recently, finite and infinite Gelfand pairs have been studied in asymptotic and geometric group theory in connection with the so-called branch groups introduced by R. I. Grigorchuk in (see).

Several examples of finite Gelfand pairs, where G is a Weyl group or a Chevalley group over a finite field were studied by Delsarte, Dunkl and Stanton (see the surveys or the book by Klimyk and Vilenkin). Delsarte was motivated by applications to association schemes of coding theory, while Dunkl and Stanton were interested in applications to orthogonal polynomials and special functions. For the point of view of the theory of association schemes see the monographs by Bailey, Bannai and Ito see also the work of Takacs, on harmonic analysis on Schur algebras, that contains several applications to probability.