2024/05/06 by Nathan Hayford, Hayford, Nathan · 3 citations
Physics and Astronomy · #Complex Network Analysis Techniques #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Mathematical Physics (math-ph) #Opinion Dynamics and Social Influence #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2405.03260
openalex publication_date 2024/05/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study a higher-order Painlevé-type equation, arising as a string equation of the 3rd 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 ℙ1. We also identify the (nonautonomous) Hamiltonian structure associated to this equation, and write a suitable τ-differential for this system. This τ-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 τ-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).