Random Overlap Structures: Properties and Applications to Spin Glasses

TL;DR

This paper studies Random Overlap Structures (ROSt's) in spin glasses, proving their stability and properties that support the ultrametricity conjecture and providing a constructive proof of the Parisi formula assuming ultrametricity.

Contribution

It establishes the continuity of the cavity mapping on ROSt's, proves stochastic stability for Gibbs measures, and offers a framework linking ultrametricity to the Parisi formula.

Findings

Proved the cavity mapping is continuous on ROSt's.

Established stochastic stability of limiting Gibbs measures.

Provided a constructive approach to the Parisi formula assuming ultrametricity.

Abstract

Random Overlap Structures (ROSt's) are random elements on the space of probability measures on the unit ball of a Hilbert space, where two measures are identified if they differ by an isometry. In spin glasses, they arise as natural limits of Gibbs measures under the appropriate algebra of functions. We prove that the so called `cavity mapping' on the space of ROSt's is continuous, leading to a proof of the stochastic stability conjecture for the limiting Gibbs measures of a large class of spin glass models. Similar arguments yield the proofs of a number of other properties of ROSt's that may be useful in future attempts at proving the ultrametricity conjecture. Lastly, assuming that the ultrametricity conjecture holds, the setup yields a constructive proof of the Parisi formula for the free energy of the Sherrington-Kirkpatrick model by making rigorous a heuristic of Aizenman, Sims and Starr.