Dimers on Riemann surfaces I: Temperleyan forests

Understand

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 topolgical 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.