The Markov property for $φ^4_3$ on the cylinder
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arXiv
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| Format: | Preprint |
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2025
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| author | Barashkov, Nikolay Gunaratnam, Trishen S. |
| author_facet | Barashkov, Nikolay Gunaratnam, Trishen S. |
| contents | We prove that the $φ^4_3$ 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 $φ^4_3$ 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 $φ^4_3$ models with rough boundary conditions. We heavily use the variational approach to $φ^4_3$ models introduced in Barashkov and Gubinelli (Duke, 2020) that is based on the Boué-Dupuis formula and dual to Polchinski's continuous renormalization group. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_21466 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Markov property for $φ^4_3$ on the cylinder Barashkov, Nikolay Gunaratnam, Trishen S. Probability Mathematical Physics 81T08, 60H30 We prove that the $φ^4_3$ 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 $φ^4_3$ 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 $φ^4_3$ models with rough boundary conditions. We heavily use the variational approach to $φ^4_3$ models introduced in Barashkov and Gubinelli (Duke, 2020) that is based on the Boué-Dupuis formula and dual to Polchinski's continuous renormalization group. |
| title | The Markov property for $φ^4_3$ on the cylinder |
| topic | Probability Mathematical Physics 81T08, 60H30 |
| url | https://arxiv.org/abs/2506.21466 |