Homotopy and holonomy of the planar Brownian motion in a Poisson punctured plane

TL;DR

This paper introduces models of random connections on principal bundles over the plane with singular curvatures, analyzing their limits and the convergence of holonomy along Brownian paths towards explicit limits.

Contribution

It develops diffeomorphism-invariant models of random connections with singular curvatures and studies their asymptotic behavior, including convergence to Yang--Mills fields and explicit holonomy limits.

Findings

Model converges to a Yang--Mills field in a specific limit.

Holonomy along Brownian trajectories converges to an explicit limit.

Provides a new framework for studying random connections with singularities.

Abstract

We define a family of diffeomorphism-invariant models of random connections on principal $G$-bundles over the plane, whose curvatures are concentrated on singular points. In a limit when the number of point grows whilst the singular curvature on each point diminishes, the model converges in some sense towards a Yang--Mills field. We study another regime for which we prove that the holonomy along a Brownian trajectory converges towards an explicit limit.