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 Wästlund (Annals of Mathematics, 2012). This fills the last remaining gap in the proof of the original Mézard-Parisi prediction for this problem (Journal de Physique Lettres, 1985).
"The Mézard-Parisi equation for matchings in pseudo-dimension $d>1$." Electron. Commun. Probab. 20 1 - 7, 2015. https://doi.org/10.1214/ECP.v20-3791