Work

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

Year: 2025

Type: article

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.

Source: Symmetry Integrability and Geometry Methods and Applications

DOI: 10.3842/sigma.2025.079

Authors: Maurice Duits, Nathan Hayford, Seung-Yeop Lee

Institutions: KTH Royal Institute of Technology, University of South Florida

Language: English

Topic: Random Matrices and Applications

Subfield: Statistics and Probability

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

Funders: Vetenskapsrådet, H2020 European Research Council

Awards: Patterns in Random Tilings, Patterns in Random Tilings