The Ruelle Probability Cascades

Abstract

In this chapter we will describe a remarkable family of random measures on a Hilbert space, called the Ruelle probability cascades, that play a central role in the Sherrington–Kirkpatrick model, and the first three sections will be devoted to the construction of these measures and study of their properties. We will see that they satisfy certain special invariance properties, one of which, called the Ghirlanda–Guerra identities, will serve as a key link between the Ruelle probability cascades and the Gibbs measure in the Sherrington–Kirkpatrick model. This connection will be explained in the last two sections, where it will be shown that, in a certain sense, the Ghirlanda–Guerra identities completely determine a random measure up to a functional order parameter. We will see in the next chapter that, as a consequence, the asymptotic Gibbs measures in the Sherrington–Kirkpatrick and mixed p-spin models can be approximated by the Ruelle probability cascades.KeywordsHilbert SpacePoisson ProcessInvariance PropertyRandom MeasureGibbs MeasureThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.