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

TL;DR

This paper calculates the genus zero free energy for a 2-matrix model related to the Ising model on random planar graphs, using advanced mathematical techniques to confirm and extend previous predictions.

Contribution

It provides a new parametric formula for the free energy and characterizes the phase space, advancing the mathematical understanding of the model's behavior.

Findings

Confirmed the genus 0 free energy formula for the Ising model on random graphs.

Developed a new parametric representation of the free energy.

Analyzed the spectral curve to facilitate future multicritical point studies.

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.