The Minimum Perfect Matching in Pseudo-dimension $0<q<1$

Abstract

It is known that for $K_{n,n}$ equipped with i.i.d. $exp(1)$ edge costs, the minimum total cost of a perfect matching converges to $\pi^2/6$ in probability. Similar convergence has been established for all edge cost distributions of pseudo-dimension $q \geq 1$, such as Wei(1,q) costs. In this paper we extend those results all $q>0$, confirming the M\'ezard-Parisi conjecture in the last remaining applicable case.