The Parisi formula completed

TL;DR

This paper completes the Parisi formula for the SK model's free energy by deriving a closed-form functional, providing a new integral representation, and establishing the stability of the solution without using the replica trick.

Contribution

It introduces a closed-form generating functional for the Parisi formula and demonstrates the stability of the solution through direct analysis.

Findings

Derived an integral representation for the Parisi differential equation solution

Expressed the free energy as a functional of order parameters

Proved the marginal stability of the thermodynamic state

Abstract

The Parisi formula for the free energy of the Sherrington-Kirkpatrick model is completed to a closed-form generating functional. We first find an integral representation for a solution of the Parisi differential equation and represent the free energy as a functional of order parameters. Then we set stationarity equations for local maxima of the free energy determining the order-parameter function on interval $[0,1]$. Finally we show without resorting to the replica trick that the solution of the stationarity equations leads to a marginally stable thermodynamic state.