An Elementary Proof of Phase Transition in the Planar XY Model

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Abstract:

Using elementary methods we obtain a power-law lower bound on the two-point function of the planar XY spin model at low temperatures. This was famously first rigorously obtained by Fröhlich and Spencer and establishes a Berezinskii-Kosterlitz-Thouless phase transition in the model. Our argument relies on a new loop representation of spin correlations, a recent result of Lammers on delocalisation of integer-valued height functions, and classical correlation inequalities. This is joint work with Diederik van Engelenburg.

Biography:

Marcin Lis is a non-tenure-track assistant professor in mathematics at the University of Vienna.

Previously, he was a research associate in the Statistical Laboratory at the University of Cambridge, Chalmers University and the University of Gothenburg, and at ICERM, Brown University. He obtained his PhD from the VU University Amsterdam.

He is mainly interested in two-dimensional statistical mechanics with a focus on the Ising model, the dimer model, the six-vertex model and random walk loop soups. He studies the associated combinatorial structures in order to better understand the critical behaviour and phase
transitions of these models.

Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai