Partial results on the convexity of the Parisi functional with PDE approach

TL;DR

This paper explores the convexity of the Parisi functional in the Sherrington-Kirkpatrick model using a PDE approach, simplifying previous probabilistic proofs and extending the results.

Contribution

It introduces a PDE-based method to analyze the convexity of the Parisi functional, providing a more general and simplified proof compared to prior probabilistic approaches.

Findings

Convexity along one-sided directions established

PDE approach simplifies the proof process

Results extend previous convexity findings

Abstract

We investigate the convexity problem for the Parisi functional defined on the space of the so-called functional ordered parameters in the Sherrington-Kirkpatrick model. In the recent work of Panchenko [3], he proved that this functional is convex along one-sided directions with a probabilistic method. In this paper, we will study this problem with a PDE approach that simplifies his original proof and presents more general results.