The Mezard-Parisi equation for matchings in pseudo-dimension d>1

TL;DR

This paper proves the existence and uniqueness of solutions to the cavity equation for the random assignment problem in pseudo-dimension greater than one, confirming a long-standing conjecture and completing the proof of the Me9zard-Parisi prediction.

Contribution

It establishes the missing proof of the Me9zard-Parisi equation solution for pseudo-dimension d>1, confirming theoretical predictions in combinatorial optimization.

Findings

Proves existence and uniqueness of the cavity equation solution for d>1

Fills the last gap in the Me9zard-Parisi prediction proof

Confirms conjectures by Aldous, Bandyopadhyay, and We4stlund

Abstract

We establish existence and uniqueness of the solution to the cavity equation for the random assignment problem in pseudo-dimension $d>1$, as conjectured by Aldous and Bandyopadhyay (Annals of Applied Probability, 2005) and W\"astlund (Annals of Mathematics, 2012). This fills the last remaining gap in the proof of the original M\'ezard-Parisi prediction for this problem (Journal de Physique Lettres, 1985).