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Ising model with external magnetic field on random planar maps: Critical exponents

Nicolas Tokka

TL;DR

This work analyzes the Ising model on random tetravalent planar maps in the presence of a weak external magnetic field. By leveraging analytic combinatorics and a Lagrangian parametrization of the generating function, it accurately characterizes the singular structure of the partition function and derives explicit expressions for spontaneous magnetization and susceptibility. The study confirms a third-order phase transition in zero field and computes the critical exponents $α=-1$, $β=\tfrac{1}{2}$, $γ=2$, and $δ=5$, matching the Boulatov-Kazakov predictions within this combinatorial setting. The results deepen the connection between random geometry and statistical mechanics, providing rigorous, enumerative insight into Ising models on random planar lattices and their scaling behavior near criticality.

Abstract

We study the Ising model with an external magnetic field on random tetravalent planar maps and investigate its critical behavior. Explicit expressions for spontaneous magnetization and the susceptibility are computed and the critical exponents $α=-1$ (third order phase transition), $β=\frac{1}{2}$ (spontaneous magnetization), $γ=2$ (susceptibility at zero external magnetic field) and $δ=5$ (magnetization at critical temperature) are derived. To do so, we study the asymptotic behavior of the partition function of the model in the case of a weak external magnetic field using analytic combinatorics.

Ising model with external magnetic field on random planar maps: Critical exponents

TL;DR

This work analyzes the Ising model on random tetravalent planar maps in the presence of a weak external magnetic field. By leveraging analytic combinatorics and a Lagrangian parametrization of the generating function, it accurately characterizes the singular structure of the partition function and derives explicit expressions for spontaneous magnetization and susceptibility. The study confirms a third-order phase transition in zero field and computes the critical exponents , , , and , matching the Boulatov-Kazakov predictions within this combinatorial setting. The results deepen the connection between random geometry and statistical mechanics, providing rigorous, enumerative insight into Ising models on random planar lattices and their scaling behavior near criticality.

Abstract

We study the Ising model with an external magnetic field on random tetravalent planar maps and investigate its critical behavior. Explicit expressions for spontaneous magnetization and the susceptibility are computed and the critical exponents (third order phase transition), (spontaneous magnetization), (susceptibility at zero external magnetic field) and (magnetization at critical temperature) are derived. To do so, we study the asymptotic behavior of the partition function of the model in the case of a weak external magnetic field using analytic combinatorics.

Paper Structure

This paper contains 8 sections, 12 theorems, 56 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Planar maps with spins : definitions and generating series
  4. Planar maps with spins : Lagrangian parametrization
  5. Ising model without external magnetic field: c=1
  6. Ising model with external magnetic field: c<=1
  7. Enumerative results and asymptotic behavior
  8. Critical exponents

Key Result

Theorem 1

For any $\nu>0$, there exists $0<\varepsilon_\nu<1$ such that for all $c\in\left[1-\varepsilon_\nu,1+\varepsilon_\nu\right]$, there exists a non-zero explicit constant $\gimel(\nu,c)$ in such a way that, as $n\rightarrow\infty$, where $\mu_{\nu,c}$ is the radius of convergence of the power series $z \mapsto \mathcal{Z}(\nu,c,z)$.

Figures (3)

  • Figure 1: The curves of the spontaneous magnetization $M_0(\nu)$ and of the susceptibility $\chi(\nu,1)$ as functions of $\nu>0$. A transition occurs at the critical point $\nu_\star=4$.
  • Figure 2: A tetravalent rooted planar map endowed with a spin configuration, whose weight is $\nu^{12}c^{3}z^{11}$ in the generating function $\mathcal{Z}$. The frustrated edges are indicated with dotted lines.
  • Figure 3: The unrooted tetravalent planar maps endowed with a spin configuration having one or two vertices. The possible rootings are given by the grey orientations. The frustrated edges are indicated by dotted lines.

Theorems & Definitions (21)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6: Theorem 5.3, AMT-Bij
  • Lemma 7
  • proof
  • Corollary 8
  • proof
  • ...and 11 more