Filling random cycles

TL;DR

This paper analyzes the asymptotic filling volume of random Lipschitz cycles in geometric spaces, extending classical combinatorial theorems and aiming to inform the topology of random links and maps.

Contribution

It generalizes the Ajtai--Komlós--Tusnády theorem to random geometric cycles, providing new estimates for minimal surfaces in random knot models.

Findings

Asymptotic estimates for filling volumes of random cycles

Extension of classical matching theorems to geometric probability

Potential applications to topology of random links and maps

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--Koml\'os--Tusn\'ady 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.