The Ising Model Coupled to 2D Gravity: Higher-order Painlev\'{e} Equations/The $(3,4)$ String Equation

TL;DR

This paper investigates a higher-order Painlevé equation linked to the Ising model coupled with 2D gravity, characterizing it through isomonodromic deformations, Hamiltonian structures, and tau-functions, revealing new insights into its integrability and properties.

Contribution

It introduces a novel higher-order Painlevé-type equation from the KP hierarchy reduction, characterizes it via isomonodromic deformations, and develops a general tau-differential formula for resonant connections.

Findings

Characterization of the equation through isomonodromic deformations

Identification of the associated Hamiltonian structure

Development of a simplified tau-differential formula

Abstract

We study a higher-order Painlev\'{e}-type equation, arising as a string equation of the $3^{rd}$ order reduction of the KP hierarchy. This equation appears at the multi-critical point of the $2$-matrix model with quartic interactions, and describes the Ising phase transition coupled to 2D gravity, cf. [1]. We characterize this equation in terms of the isomonodromic deformations of a particular rational connection on $\mathbb{P}^{1}$. We also identify the (nonautonomous) Hamiltonian structure associated to this equation, and write a suitable $\tau$-differential for this system. This $\tau$-differential can be extended to the canonical coordinates of the associated Hamiltonian system, allowing us to verify Conjectures 1. and 2. of [2]. We also present a fairly general formula for the $\tau$-differential of a special class of resonant connections, which is somewhat simpler than that of [3]. [1] M. Duits, N. Hayford, and S.-Y. Lee. "The Ising Model Coupled to 2D Gravity: Genus Zero Partition Function". arXiv preprint, 2023. [2] A.R. Its and A. Prokhorov. "On some Hamiltonian properties of the isomonodromic tau functions". Rev. Math. Phys. 30.7 (2018). [3] M. Bertola and M.Y. Mo. "Isomonodromic deformation of resonant rational connections". Int. Math. Res. Pap. 11 (2005).