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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2511.06321 |
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| _version_ | 1866918192342892544 |
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| author | Park, Jiwoon |
| author_facet | Park, Jiwoon |
| contents | The $n$-component weakly coupled $|φ|^4$ model on the $\Z^d$ lattice ($d\ge 4$) exhibits a critical two-point correlation function with an exact polynomial decay in infinite volume, regardless of whether the interaction is short- or long-range. This paper presents a rigorous analysis of the system in both $\Z^d$ and a finite-volume torus. In a torus, we prove the existence of a plateau effect, where the correlation function undergoes a crossover from the polynomial decay to a uniform constant state.
We then establish the precise scaling limit picture that provides a complementary description of this crossover. As immediate consequences, we verify the finite-size scaling limit predicted by Zinn-Justin, the finite-size scaling exponents (qoppas) suggested by Kenna and Berche and the role of the Fourier modes in finite-size scaling suggested by Flores-Sola, Berche, Kenna and Weigel. The proofs use the renormalisation group map constructed in the author's previous work. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_06321 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Torus scaling limits and the plateau of the critical weakly coupled $|φ|^4$ model in $d \ge 4$ Park, Jiwoon Probability Mathematical Physics The $n$-component weakly coupled $|φ|^4$ model on the $\Z^d$ lattice ($d\ge 4$) exhibits a critical two-point correlation function with an exact polynomial decay in infinite volume, regardless of whether the interaction is short- or long-range. This paper presents a rigorous analysis of the system in both $\Z^d$ and a finite-volume torus. In a torus, we prove the existence of a plateau effect, where the correlation function undergoes a crossover from the polynomial decay to a uniform constant state. We then establish the precise scaling limit picture that provides a complementary description of this crossover. As immediate consequences, we verify the finite-size scaling limit predicted by Zinn-Justin, the finite-size scaling exponents (qoppas) suggested by Kenna and Berche and the role of the Fourier modes in finite-size scaling suggested by Flores-Sola, Berche, Kenna and Weigel. The proofs use the renormalisation group map constructed in the author's previous work. |
| title | Torus scaling limits and the plateau of the critical weakly coupled $|φ|^4$ model in $d \ge 4$ |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2511.06321 |