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

TL;DR

This paper establishes the Markov property for the $^4_3$ quantum field model on cylinders, characterizes its boundary laws, and introduces a probabilistic approach to analyze its spectral properties and Hamiltonian construction.

Contribution

It proves the Markov property for $^4_3$ on cylinders, characterizes boundary laws, and offers an alternative probabilistic construction of the Hamiltonian with new spectral insights.

Findings

Proves $^4_3$ satisfies a Markov property on cylinders.

Characterizes boundary laws up to absolutely continuous perturbations.

Shows the Hamiltonian has discrete spectrum and a Perron-Frobenius type ground state result.

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 Bou\'e-Dupuis formula and dual to Polchinski's continuous renormalization group.