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Nathan Hayford: The Ising Model Coupled to 2D Gravity

Time: Tue 2022-11-29 15.15 - 16.15

Location: Zoom

Video link: Meeting ID: 698 3346 0369

Participating: Nathan Hayford (University of South Florida)

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One of the most celebrated exactly solvable models in statistical mechanics is the two-dimensional Ising model. The original model, introduced in the 1920s, has a rich mathematical structure. It thus came as a pleasant surprise when physicists studying matrix models of 2D gravity found that, coupled to quantum gravity, the planar Ising model still had an elegant solution. The methods used by V. Kazakov and his collaborators involved the method of orthogonal polynomials. However, these methods were formal, and no direct analytic derivation of the phase transition has been described in the literature since the original paper of V. Kazakov in 1986. In this talk, we present a rigorous proof of Kazakov’s results, using steepest descent analysis for biorthogonal polynomials. We are able to calculate the genus 0 partition function, and we also find that the phase transition is described by the string equation of a 3rd order reduction of the KP hierarchy, in agreement with the predictions of G. Moore, M. Douglas, and their collaborators. This is joint work with Maurice Duits and Seung-Yeop Lee.