Abstract

We review old and new uses of exchangeability, emphasizing the general theme of exchangeable representations of complex random structures. Illustrations of this theme include processes of stochastic coalescence and fragmentation; continuum random trees; second-order limits of distances in random graphs; isometry classes of metric spaces with probability measures; limits of dense random graphs; and more sophisticated uses in finitary combinatorics.

AMS subject classification (MSC2010) 60G09, 60C05, 05C80

Introduction

Kingman's write-up of his 1977 Wald Lectures drew attention to the subject of exchangeability, and further indication of the topics of interest around that time can be seen in the write-up of my 1983 Saint-Flour lectures. As with any mathematical subject, one might expect some topics subsequently to wither, some to blossom and new topics to emerge unanticipated. This Festschrift paper aims, in informal lecture style,

1. (a) to recall the state of affairs 25 years ago (sections 2.1–2.3, 3.1);

2. (b) to briefly describe three directions of subsequent development that have recently featured prominently in monographs (sections 2.4, 3.1–3.2);

3. (c) to describe 3 recent rediscoveries, motivated by new theoretical topics outside mainstream mathematical probability, of the theory of representations of partially exchangeable arrays (sections 2.5, 5.1–5.2);

4. (d) to emphasize a general program that has interested me for 20 years. It doesn't have a standard name, but let me here call it exchangeable representations of complex random structures (section 4).