# Holonomy of the Planar Brownian Motion in a Poisson Punctured Plane

## TL;DR

This paper explores how random connections behave on a plane with many small singular points, showing that their holonomy converges to a specific limit.

## Contribution

The paper introduces new diffeomorphism-invariant models of random connections and proves convergence of holonomy in a Poisson punctured plane.

## Key findings

- A family of diffeomorphism-invariant random connections is defined with curvature concentrated on singular points.
- The holonomy along a Brownian trajectory converges to an explicit limit as singular points increase and curvature diminishes.
- The limit behavior is derived when the number of singular points grows to infinity.

## 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. We study the limit when the number of points grows to infinity whilst the singular curvature on each point diminishes, and prove that the holonomy along a Brownian trajectory converges towards an explicit limit.