Asymptotic Properties of a Special Solution to the (3,4) String Equation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911227444199424 |
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| author | Hayford, Nathan |
| author_facet | Hayford, Nathan |
| contents | We analyze the asymptotic properties a special solution of the $(3,4)$ string equation, which appears in the study of the multicritical quartic $2$-matrix model. In particular, we show that in a certain parameter regime, the corresponding $τ$-function has an asymptotic expansion which is `topological' in nature. Consequently, we show that this solution to the string equation with a specific set of Stokes data exists, at least asymptotically. We also demonstrate that, along specific curves in the parameter space, this $τ$-function degenerates to the $τ$-function for a tritronquée solution of Painlevé I (which appears in the critical quartic $1$-matrix model), indicating that there is a `renormalization group flow' between these critical points. This confirms a conjecture from [1].
[1] The Ising model, the Yang-Lee edge singularity, and 2D quantum gravity, C. Crnković, P. Ginsparg, G. Moore. Phys. Lett. B 237 2 (1990) |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_22646 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotic Properties of a Special Solution to the (3,4) String Equation Hayford, Nathan Complex Variables Mathematical Physics Exactly Solvable and Integrable Systems We analyze the asymptotic properties a special solution of the $(3,4)$ string equation, which appears in the study of the multicritical quartic $2$-matrix model. In particular, we show that in a certain parameter regime, the corresponding $τ$-function has an asymptotic expansion which is `topological' in nature. Consequently, we show that this solution to the string equation with a specific set of Stokes data exists, at least asymptotically. We also demonstrate that, along specific curves in the parameter space, this $τ$-function degenerates to the $τ$-function for a tritronquée solution of Painlevé I (which appears in the critical quartic $1$-matrix model), indicating that there is a `renormalization group flow' between these critical points. This confirms a conjecture from [1]. [1] The Ising model, the Yang-Lee edge singularity, and 2D quantum gravity, C. Crnković, P. Ginsparg, G. Moore. Phys. Lett. B 237 2 (1990) |
| title | Asymptotic Properties of a Special Solution to the (3,4) String Equation |
| topic | Complex Variables Mathematical Physics Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2507.22646 |