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Nathan Hayford

Publications and source records attributed to Nathan Hayford.

8 recordsLinked to original sources
arXiv2026

Universal Asymptotics and Exact Enumeration of Eulerian Maps

We calculate the asymptotics of the number of connected, labeled, genus $g$ Eulerian maps with an arbitrary degree sequence, in the limit as the total number of vertices tends to infinity. This asymptotic is universal, in the sense that the leading order term depends on only finitely many map characteristics. The constant factor in this formula is related to the Painlev\'{e} I equation. Our methods combine the analysis of the recurrence coefficients associated to a particular family of orthogonal polynomials, and the theory of analytic combinatorics of several variables. We also derive an exact formula for the number of connected, labeled, genus $1$ Eulerian maps. These are the first results on this kind of enumeration problem for $g\geq 1$, non-regular (mixed-valence) maps.

math.CO↗
arXiv2026

The Ising Model Coupled to 2D Gravity: Critical Partition Function

We prove that the differential of the log of the partition function for the $2$-matrix model with quartic interactions converges in a certain double-scaling regime to the differential of the $\boldsymbol{\tau}$-function for the $(3,4)$ string equation. This confirms the convergence of the critical Ising model on random surfaces to the $(3,4)$ topological minimal model, which was stated in the works of Douglas and Shenker, Br\'{e}zin and Kazakov, and Gross and Migdal. Our analysis is based on a steepest-descent analysis of a Riemann-Hilbert problem associated to a family of biorthogonal polynomials. New features in the matching problem in the construction of local parametrices appear.

math-ph↗
arXiv2025

The Painlev\'{e} I hierarchy: Correspondence between the isomonodromic approach and the minimal models of the KP hierarchy

Two approaches to the Painlev\'{e} I hierarchy are discussed: the isomonodromic construction based on meromorphic connections, and the minimal models construction based on a reduction of the KP hierarchy. An explicit correspondence between both formalisms is built which gives the identification of these setups. In particular, this provides new expressions for the Lax matrices and Hamiltonians.

math-ph↗
arXiv2025

Asymptotic Properties of a Special Solution to the (3,4) String Equation

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 $\tau$-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 $\tau$-function degenerates to the $\tau$-function for a tritronqu\'{e}e solution of Painlev\'{e} 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. Crnkovi\'{c}, P. Ginsparg, G. Moore. Phys. Lett. B 237 2 (1990)

math.CV↗
arXiv2024

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

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).

math-ph↗
arXiv2024

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

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.

math-ph↗
arXiv2021

A Note on Electrified Droplets

We give an in-depth analysis of a 1-parameter family of electrified droplets first described in D. Khavinson et. al. (2005). We also investigate a technique for searching for new solutions to the droplet equation, and rederive via this technique a 1-parameter family of physical droplets, which were first discovered by D. Crowdy (1999). We speculate on extensions of these solutions, in particular to the case of a droplet with multiple connected components.

math.CV↗