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Séminaire de Probabilités commun ICJ/UMPA

Dmitry Chelkak, "Double-dimers and CLE(4): what is still missing after Kenyon and Dubédat?"

jeudi 26 octobre 2017 de au (Europe/Paris)
at UMPA ( salle 435 )
Description
It is a well known prediction that, at least in the case of Temperley discretizations, individual double-dimer interfaces and the full loop ensemble converge to SLE(4) curves and the nested CLE(4), respectively. After the breakthrough works of Kenyon (the link between statistics of loops and determinants of vector-bundle Laplacians defined by SL(2,C) connections) and of Dubédat (the link with tau-functions in the case of locally unipotent monodromies), the current state-of-the-art looks like that this prediction is "almost proven". Nevertheless, there are some nontrivial "technicalities" needed to complete such a virtual proof, thus the main goal of this _informal_ talk is to discuss what is still missing, especially on the probabilistic side. In particular, it seems that a good RSW-type theory for double-dimer interfaces should be eventually sufficient to complete the argument (warning: this is not about the topology of convergence). On a more rigorous note, I will explain how to derive the convergence of the probability that a double-dimer interface in a Temperley domain (linking two marked boundary points) passes to the left of two given inner points from a version of Dubédat's results. Based on a joint project with Mikhail Basok (St.Petersburg).