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arXiv · 2606.27125

The Ising Model Coupled to 2D Gravity: Critical Partition Function

Maurice Duits·Nathan Hayford·Seung-Yeop Lee
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Abstract

We prove that the differential of the log of the partition function for the $2$-matrix model with quartic interactions converges in a certain double-scaling regime to the differential of the $\boldsymbol{\tau}$-function for the $(3,4)$ string equation. This confirms the convergence of the critical Ising model on random surfaces to the $(3,4)$ topological minimal model, which was stated in the works of Douglas and Shenker, Br\'{e}zin and Kazakov, and Gross and Migdal. Our analysis is based on a steepest-descent analysis of a Riemann-Hilbert problem associated to a family of biorthogonal polynomials. New features in the matching problem in the construction of local parametrices appear.

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Maurice Duits, Nathan Hayford, Seung-Yeop Lee. 2026-06-25. The Ising Model Coupled to 2D Gravity: Critical Partition Function. https://arxiv.org/abs/2606.27125

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