An online afternoon workshop on 'Dimers and related models' was held on 22 September 2021. Organised by Sunil Chhita and Alex Watson for the RSS Applied Probability Section, it featured three talks, given respectively by Sanjay Ramassamy (CEA Saclay), Qian Wei (Université Paris-Saclay) and István Prause (University of Eastern Finland).
The dimer model originates from statistical mechanics. One of the nice features of the dimer model is that it is exactly solvable, which has made it a very useful surrogate model to study phase transitions. These phase transitions arise everywhere in nature and are very complex, so to study them analytically is incredibly useful. One fascinating aspect of the dimer model is its connections to other areas of mathematics, such as algebra and geometry.
The first talk of the afternoon was given by Sanjay Ramassamy, which described a correspondence between the dimer model on bipartite planar graphs and embeddings arising naturally from centres of circle patterns. Sanjay introduced a set of local transformations where the correspondence still holds and is also an example of a discrete integrable system, where the embeddings are harmonic. He then showed that this indicates that one should expect to see conformally invariant scaling limits.
Next up was Qian Wei, who talked on a particular dimer model with a ‘free boundary’. After showing methods to compute entries of the inverse Kasteleyn matrix, which gives insights into the correlations behind the model, Wei showed that the limiting height function fluctuations are given by Gaussian Free field with Neumann Boundary Conditions. This settled a question posed by Giuliani, Jauslin and Lieb.
The final talk of the day was given by István Prause, who talked on the five-vertex model, a type of non-intersecting lattice path model which is closely related to the dimer model. It is known that the correlations for this model are non-determinantal, so studying this model is inherently more difficult than the dimer model. Even without having determinantal structure, István showed that it is still possible to obtain the exact phase diagram, free energy and the surface tension for the model, which gives key insights into the behaviour in the scaling limit.