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

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