Dimers on Riemann surfaces I: Temperleyan forests

This is the first article in a series of two papers in which we study the Temperleyan dimer model on an arbitrary bounded Riemann surface of finite topological type. The end goal of both papers is to prove the convergence of height fluctuations to a universal and conformally invariant scaling limit. In this part, we show that the dimer model on the Temperleyan superposition of a graph embedded on the surface and its dual is well posed, provided that we remove an appropriate number of punctures. We further show that the resulting dimer configuration is in bijection with an object which we call Temperleyan forest, whose law is characterised in terms of a certain topological condition. Finally, we discuss the relation between height differences and Temperleyan forest and prove that the convergence of the latter (which is the subject of the second paper in this series) implies the existence of a conformally invariant scaling limit of height fluctuations.