Abstract:
Livine and Bonzom recently proposed a geometric formula for a certain set of complex zeros of the partition function of the Ising model defined on planar graphs. Remarkably, the zeros depend locally on the geometry of an immersion of the graph in the three dimensional Euclidean space (different immersions give rise to different zeros). When restricted to the flat case, the weights become the critical weights on circle patterns. We rigorously prove the formula by geometrically constructing a null eigenvector of the Kac-Ward matrix whose determinant is the squared partition function. The main ingredient of the proof is the realisation that the associated Kac-Ward transition matrix gives rise to an SU(2) connection on the graph, creating a direct link with rotations in three dimensions. The existence of a null eigenvector turns out to be equivalent to this connection being flat.
Biography:
Marcin Lis is an assistant professor in mathematics at the Vienna University of Technology. Previously he was a research associate at the University of Vienna, 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 spin and height-function models. He studies the associated combinatorial structures in order to better understand the critical behaviour and phase transitions of these models.
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