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The Markov property for $\varphi^4_3$ on the cylinder

Nikolay Barashkov, Trishen S. Gunaratnam

TL;DR

This work advances Euclidean quantum field theory by proving Segal-like gluing and a Markov property for the φ^4_3 model on cylinders and tori, even when the boundary law is singular relative to the boundary GFF. The authors develop a novel variational framework (Boué-Dupuis) combined with paracontrolled calculus to construct φ^4_3 models with rough boundary conditions, identify and renormalize boundary and bulk divergences, and establish a limiting boundary measure ν^0 governing gluing. They then build a φ^4_3 Hamiltonian on the 2D torus, proving discreteness of spectrum, simple ground state, and positivity, with eigenfunctions lying in Orlicz spaces. The results demonstrate Segal-type gluing for non-Gaussian boundary data, offering a probabilistic path to spectral properties and potentially extending reconstruction theorems to the full space. Together, these contributions provide a rigorous bridge between probabilistic boundary theory and quantum field theoretical structures in a high-dimensional, singular setting.

Abstract

We prove that the $\varphi^4_3$ model satisfies a version of Segal's axioms in the special case of three-dimensional tori and cylinders. As a consequence, we give the first proof that this model satisfies a Markov property and we characterize its boundary law up to absolutely continuous perturbations. In addition, we use Segal's axioms to give an alternative construction of the $\varphi^4_3$ Hamiltonian on two-dimensional tori as compared with Glimm (Comm. Math. Phys., 1968). We exploit this probabilistic approach to prove novel fundamental spectral properties of the Hamiltonian, such as discrete spectrum and a Perron-Froebenius type result on its ground state. The key technical contributions of this article are the development of tools to analyze $\varphi^4_3$ models with rough boundary conditions. We heavily use the variational approach to $\varphi^4_3$ models introduced in Barashkov and Gubinelli (Duke, 2020) that is based on the Boué-Dupuis formula and dual to Polchinski's continuous renormalization group.

The Markov property for $\varphi^4_3$ on the cylinder

TL;DR

This work advances Euclidean quantum field theory by proving Segal-like gluing and a Markov property for the φ^4_3 model on cylinders and tori, even when the boundary law is singular relative to the boundary GFF. The authors develop a novel variational framework (Boué-Dupuis) combined with paracontrolled calculus to construct φ^4_3 models with rough boundary conditions, identify and renormalize boundary and bulk divergences, and establish a limiting boundary measure ν^0 governing gluing. They then build a φ^4_3 Hamiltonian on the 2D torus, proving discreteness of spectrum, simple ground state, and positivity, with eigenfunctions lying in Orlicz spaces. The results demonstrate Segal-type gluing for non-Gaussian boundary data, offering a probabilistic path to spectral properties and potentially extending reconstruction theorems to the full space. Together, these contributions provide a rigorous bridge between probabilistic boundary theory and quantum field theoretical structures in a high-dimensional, singular setting.

Abstract

We prove that the model satisfies a version of Segal's axioms in the special case of three-dimensional tori and cylinders. As a consequence, we give the first proof that this model satisfies a Markov property and we characterize its boundary law up to absolutely continuous perturbations. In addition, we use Segal's axioms to give an alternative construction of the Hamiltonian on two-dimensional tori as compared with Glimm (Comm. Math. Phys., 1968). We exploit this probabilistic approach to prove novel fundamental spectral properties of the Hamiltonian, such as discrete spectrum and a Perron-Froebenius type result on its ground state. The key technical contributions of this article are the development of tools to analyze models with rough boundary conditions. We heavily use the variational approach to models introduced in Barashkov and Gubinelli (Duke, 2020) that is based on the Boué-Dupuis formula and dual to Polchinski's continuous renormalization group.

Paper Structure

This paper contains 77 sections, 96 theorems, 658 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Construction of $\varphi^4_3$ models with rough boundary conditions
  3. Gluing of amplitudes and the Markov property
  4. The $\varphi^4_3$ Hamiltonian on two-dimensional tori
  5. Strategy of proof of Theorem \ref{['thm: main']}
  6. Part I: Boundary theory
  7. Part II: Bulk amplitudes
  8. Open problems
  9. Paper organization
  10. Acknowledgements.
  11. Preliminaries
  12. Notation
  13. Gaussian Free Fields in the bulk
  14. Regularization and Wick powers in the bulk
  15. Domain Markov property and boundary GFF
  16. ...and 62 more sections

Key Result

Theorem 1.1

Let $\varphi_-,\varphi_+$ be i.i.d sampled according to an admissible boundary law $\tilde{\nu}^0$. Then almost surely there exists a choice of sequence of renormalizations $(\gamma^M_T,\delta^M_T)_{T \geq 0}$ such the sequence $(\nu_T(\, \cdot \mid \varphi_-,\varphi_+))_{T \geq 0}$ converges weakly

Figures (2)

  • Figure 1: Pictorial representation of Segal's gluing property as stated in Theorem \ref{['thm: main']}. Amplitudes on $M^-$ and $M^+$ are glued by integrating the boundary field $\varphi^0$ according to the interacting boundary measure $\nu^0$ to produce the amplitude on $M$.
  • Figure 2: The regularized amplitudes $\mathcal{A}_T(f \mid \varphi_-,\varphi_+)$ are lifted to $\boldsymbol{\mathcal{A}}_T(f, \Xi^\partial_T(\varphi_-,\varphi_+))$. Theorems \ref{['thm: bulk']} and \ref{['thm: enhancement converge']} yield convergence of these lifted amplitudes to $\boldsymbol{\mathcal{A}}_\infty(f, \Xi^\partial_T(\varphi_-,\varphi_+))$, that we then define to be the $\varphi^4_3$ amplitude.

Theorems & Definitions (202)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 2.1
  • Remark 2.2
  • Proposition 2.3
  • ...and 192 more