The goal of this post is to give an upper bound of the free energy density of the Sherrington-Kirkpatrick model using the Guerra-RSB interpolation. In order to do this, however, we will need to introduce a rewrite of the Paris formula over the Ruelle Probability Cascades (RPCs). To do this, we will first formally introduce the construction of the RPCs, some of its fundamental properties, the rewrite of the Parisi formula over them, and then the use of Gaussian interpolation from the SK model to the RPCs decorated with certain Gaussian processes leading to the upper bound.

Table of Contents

1. The Ruelle Probability Cascades
   • Poisson-Dirichlet Processes
   • Ultrametricity
   • RPC Averages and Gaussian Process Decorations
2. The Guerra RSB Bound
   • The interpolation
   • Upper bound via Convexity