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arXiv · 2405.03259

The Ising Model Coupled to 2D Gravity: Genus Zero Partition Function

Maurice Duits·Nathan Hayford·Seung-Yeop Lee
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Abstract

We compute the genus 0 free energy for the 2-matrix model with quartic interactions, which acts as a generating function for the Ising model's partition function on a random, 4-regular, planar graph. This is consistent with the predictions of Kazakov and Boulatov on this model, as well as subsequent confirmation of this formula using combinatorial methods. We also provide a new parametric formula for the free energy and give a characterization of the phase space. Our analysis is based on a steepest descent Riemann-Hilbert analysis of the associated biorthogonal polynomials and the corresponding isomonodromic $\tau$-function. A key ingredient in the analysis is a parametrization of the spectral curve. This analysis lays the groundwork for the subsequent study of the multicritical point, which we will study in a forthcoming work.

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Maurice Duits, Nathan Hayford, Seung-Yeop Lee. 2024-05-06. The Ising Model Coupled to 2D Gravity: Genus Zero Partition Function. https://doi.org/10.3842/sigma.2025.079

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