Exchangeability and continuum limits of discrete random structures

Exchangeable representations of complex random structures are useful in several ways, in particular providing a moderately general way to derive continuum limits of discrete random structures. I shall describe an old example (continuum random trees) and a more recent example (dense graph limits). Thinking this way about road routes suggests challenging new problems in the plane. Mathematics Subject Classification (2000). Primary 60G09; Secondary 60C05. This write-up follows the style of the ICM talk, presented as 5 episodes in the development of a topic over the last 80 years. • Exchangeability and de Finetti’s theorem (1930s 50s) • Structure theory for partially exchangeable arrays (1980s) • A general program for continuum limits of discrete random structures, illustrated by trees (1990s) • 3 recent “pure math” developments (2000s) • Road routes from this viewpoint (2010s) An expanded version of the material in sections 1-4 appears as a longer survey article [4]. Section 5 is work in progress, in part with Wilfrid Kendall, not yet written up in more detail. ∗Research supported by N.S.F Grant DMS-0704159. Department of Statistics, 367 Evans Hall # 3860, U.C. Berkeley CA 94720. E-mail: [email protected]