Universal fluctuation of the dimer model on a Riemann surface.

Abstract: We prove that the scaling limit of the height function fluctuation of the dimer model on a Temperleyan graph embedded on a Riemann surface  is universal, in the sense that the scaling limit exists and is conformally invariant. The result is true under the assumption that the graph satisfies an invariance principle. This extends a work of Dubedat for the torus where the limiting law was derived earlier but for isoradial graphs. Along the way we prove that the scaling limit of cycle rooted spanning forests on Riemann  surfaces converge, is universal and conformally invariant, which is of independent interest. The last results extends a work of Kassel and Kenyon.

 

Joint work with Nathanael Berestycki and Benoit Laslier.