Exchangeable random measures

TL;DR

This paper establishes a representation theorem for exchangeable random measures on array spaces indexed by subsets of natural numbers, extending classical exchangeability results and applying them to spin glass models and PSD matrices.

Contribution

It provides a new representation theorem for exchangeable random measures on array spaces, generalizing classical exchangeability theorems and applying to spin glasses and PSD matrices.

Findings

New proof of Dovbysh-Sudakov Representation Theorem

Representation theorem for exchangeable measures on array spaces

Application to limit objects in spin glass models

Abstract

Let A be a standard Borel space, and consider the space A^{\bbN^{(k)}} of A-valued arrays indexed by all size-k subsets of \bbN. This paper concerns random measures on such a space whose laws are invariant under the natural action of permutations of \bbN. The main result is a representation theorem for such `exchangeable' random measures, obtained using the classical representation theorems for exchangeable arrays due to de Finetti, Hoover, Aldous and Kallenberg. After proving this representation, two applications of exchangeable random measures are given. The first is a short new proof of the Dovbysh-Sudakov Representation Theorem for exchangeable PSD matrices. The second is in the formulation of a natural class of limit objects for dilute mean-field spin glass models, retaining more information than just the limiting Gram-de Finetti matrix used in the study of the Sherrington-Kirkpatrick model.