Speaker: Mikhail Basok (ENS Paris)

Title:
Double dimer model in Temperleyn domains and convergence to CLE(4): improvement of the topology of convergence.

Abstract:
Double dimer model is a planar statistical model constructed by sampling independently two random dimer configurations in the same planar discrete domain. Superimposing these dimer configurations we obtain a random loop ensemble associated with the model.
Consider a sequence of Temperleyn polygons on a square grid approximating a simply-connected open domain on the plane as the mesh size tends to zero, and sample the double dimer model in each of these polygons. It is expected that the sequence of the corresponding random loop ensembles converges to CLE(4) in the limit domain. Still not fully proven, this conjecture is strongly supported by breakthrough works of Richard Kenyon and Julien Dubedat where the topological observables for the double dimer model were introduced and their convergence to topological observables for CLE(4) was proven in the Temperleyn setup. It was shown by Dmitry Chelkak and the author that this result uniquely determine the limit distribution provided the limit exists; moreover, under the additional result of Tianyi Bai and Yijun Wan, this distribution was shown to coincide with CLE(4). Due to all these results, the convergence of double dimer random loop ensembles to CLE(4) in the Temperleyn setup comes down to the question whether the corresponding sequence of probability measures is tight in the topology of interest.
In this talk we will discuss certain RSW-type estimates for the double dimer model in a Temperleyn domain recently obtained by Benoit Laslier (unpublished), and introduce a topology where the convergence of the double dimer loop ensembles to CLE(4) can be finally justified from the results above. Based on a joint work with Benoit Laslier.