Speaker: Dmitry Chelkak, University of Michigan & ENS Paris
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Title: S-embeddings of planar graphs carrying the Ising model
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Abstract:
The aim of this talk is to discuss special discrete surfaces in the Minkowski space R^{2,1} and their projections to the complex plane, the so-called s-embeddings of - possibly very irregular - planar graphs carrying the nearest-neighbor Ising model. Critical Ising models on infinite doubly periodic grids admit canonical doubly periodic s-embeddings, which is the starting point of the proof of their convergence in the small mesh size limit given in arXiv:2006.14559. This universality result was not available until recently due to the lack of relevant discrete complex analysis techniques. Even more importantly, at least from our perspective, this construction provides a natural `geometric’ interpretation of the mass in non-critical Ising models and leads to an open question on the existence of the so-called `perfect’ s-embeddings of finite planar graphs carrying the Ising model into the interior of the unit hyperboloid in R^{2,1} or, equivalently, into the AdS(2,1) space with a marked point.