Exchangeable random arrays

These notes were written to accompany a sequence of workshop lectures at the Indian Institute of Science, Bangalore in January 2013. My lectures had two purposes. The first was to introduce some of the main ideas of exchangeability theory at a level suitable for graduate students (assuming familiarity with measure-theoretic probability). The second was to show the key rôle exchangeability has played in recent work on mean-field spin glass models, particularly Panchenko’s proof of a version of the Ultrametricity Conjecture. To my taste this may be the single most exciting reason to learn about exchangeability. Of course, these dual purposes constrained the choice of material: I had to focus on those varieties of exchangeability that arise in spin glass theory. Also, the spin glass results were far too complicated to cover in depth, so in those lecture I omitted almost all detail, and the notes are the same. A much more complete treatment of similar material will appear in Panchenko’s monograph [Panar]. These notes are certainly not intended to compete with that, but offer a more basic overview that may benefit the newcomer to the field.