Constructive Cavity Method

TL;DR

This work derives the Parisi functional for the Sherrington-Kirkpatrick model from the Cavity Method by assuming the equilibrium state factorizes into a product of independent Random Energy Models (REM) across an L-level Replica Symmetry Breaking (RSB) structure. A cavity-based incremental pressure $A(\\mu)$ is introduced, yielding a lower bound $p \ge \liminf_{N\to\infty} A(\\mu)$ that becomes exact in the thermodynamic limit. Under a REM-based L-RSB ansatz, the authors show that the cavity variables decompose into Gaussian-like increments and, after averaging over the REM measures via Poisson Point Process (PPP) techniques, the Parisi functional emerges as $A(\\mu) = \log 2 + \log Y_0 - \frac{\\beta^2}{4} \sum_{\\ell} \\lambda_{\\ell} (q_{\\ell}^2 - q_{\\ell-1}^2)$. They also discuss two equivalent factorization schemes that yield the same functional, highlighting the connection between cavity methods, REM universality, and ultrametric RSB kernels. Overall, the paper provides a minimal, constructive path to obtain the Parisi functional from the Cavity Method and clarifies the role of REM in RSB theory.

Abstract

We show that the functional appearing in the celebrated Parisi formula for the free energy of the Sherrington-Kirkpatrick model can be found from the incremental free energy obtained by Cavity Method if one assumes that the state is a product of independent Random Energy models.