The Ising Model Coupled to 2D Gravity: Higher-order Painlevé Equations/The $(3,4)$ String Equation
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| Format: | Preprint |
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2024
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| _version_ | 1866913894209945600 |
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| author | Hayford, Nathan |
| author_facet | Hayford, Nathan |
| contents | We study a higher-order Painlevé-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 $τ$-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). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_03260 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Ising Model Coupled to 2D Gravity: Higher-order Painlevé Equations/The $(3,4)$ String Equation Hayford, Nathan Mathematical Physics Exactly Solvable and Integrable Systems We study a higher-order Painlevé-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 $τ$-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). |
| title | The Ising Model Coupled to 2D Gravity: Higher-order Painlevé Equations/The $(3,4)$ String Equation |
| topic | Mathematical Physics Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2405.03260 |