Strict Convexity of the Parisi Formula: A Dynamic Programming Approach

Aukosh Jagannath, Ian Tobasco

Published 2015-02-16Version 1

G. Parisi predicted a novel variational formula for the thermodynamic limit of the intensive free energy in the Sherrington-Kirkpatrick model. The minimizer of this functional is predicted to be the order parameter for this system. Proving the uniqueness of this minimizer was a problem of interest to the mathematical mean field spin glasses community for some time. In this note, we present a new proof of the strict convexity of the Parisi functional using elementary techniques from stochastic dynamic programming, thereby concluding uniqueness. We also present an alternative derivation of basic properties of the Parisi PDE using standard techniques from parabolic PDEs.