Holonomy of the planar Brownian motion in a Poisson punctured plane (Q6559090)

The author defines a family of diffeomorphism-invariant models of random connections on principal \(G\)-bundles over the plane, whose curvatures are concentrated on singular points. The model presented here corresponds to a general interaction, associated with an arbitrary connected compact Lie group \(G,\) typically non-abelian. It is considered a random flat \(G\)-bundle over \(\mathbb{R}_2\setminus \mathit{P},\) each point of \(\mathit{P}\) being associated with a monodromy element whose distance to 1 in \(G\) is proportional to \(K^{-1}.\) The authors studies 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.