Filling random cycles

Abstract

We compute the asymptotic behavior of the average-case filling volume for certain models of random Lipschitz cycles in the unit cube and sphere. For example, we estimate the minimal area of a Seifert surface for a model of random knots first studied by Millett. This is a generalization of the classical Ajtai–Komlós–Tusnády optimal matching theorem from combinatorial probability. The author hopes for applications to the topology of random links, random maps between spheres, and other models of random geometric objects.