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The dimer model on Riemann surfaces

Banff International Research Station for Mathematical Innovation and Discovery

We develop a framework to study the dimer model on Temperleyan graphs embedded on a Riemann surface with finitely many holes and handles. We show that the dimer model can be understood in terms of an object which we call Temperleyan forests and show that if the Temperleyan forest has a scaling limit then the fluctuations of the height one-form of the dimer model also converge. Furthermore, if the Riemann surface is either a torus or an annulus, we show that Temperleyan forests reduce to cycle-rooted spanning forests and show convergence of the latter to a conformally invariant, universal scaling limit. As a consequence, the dimer height one-form fluctuations also converge on these surfaces, and the limit is conformally invariant. A key idea here is the geometric description of the scaling limit of a cycle-rooted spanning forest in the universal cover of the surface, achieved using tools coming in particular from the Fuschian theory of hyperbolic Riemann surfaces. Joint work with Benoit Laslier (Paris) and Gourab Ray (Victoria).

Mathematics; Probability Theory And Stochastic Processes

video/mp4

Author affiliation: University of Vienna

2020-05-20

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/74528

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/