Phase transitions for fractional $Φ^3_d$ on the torus
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915127981244416 |
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| author | Nikov, Niko |
| author_facet | Nikov, Niko |
| contents | We consider the fractional $Φ^3_d$-measure on the $d$-dimensional torus, with Gaussian free field having inverse covariance $(1-Δ)^α$, and show a phase transition at $d=3α$. More precisely, in a regular regime $d<3α$, one can construct and normalise this measure, and obtain a measure which is absolutely continuous with respect to the Gaussian free field $μ$. At $d=3α$, the behaviour depends on the size $|σ|$ of the nonlinearity: for $|σ|\ll1$, the measure exists, but is singular with respect to $μ$, whereas for $|σ|\gg1$, the measure is not normalisable. This generalises a result of Oh, Okamoto, and Tolomeo (2025) on the $Φ^3_3$-measure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_17669 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Phase transitions for fractional $Φ^3_d$ on the torus Nikov, Niko Probability Mathematical Physics Analysis of PDEs 60H30 (Primary) 81T08, 82B26, 35Q55 (Secondary) We consider the fractional $Φ^3_d$-measure on the $d$-dimensional torus, with Gaussian free field having inverse covariance $(1-Δ)^α$, and show a phase transition at $d=3α$. More precisely, in a regular regime $d<3α$, one can construct and normalise this measure, and obtain a measure which is absolutely continuous with respect to the Gaussian free field $μ$. At $d=3α$, the behaviour depends on the size $|σ|$ of the nonlinearity: for $|σ|\ll1$, the measure exists, but is singular with respect to $μ$, whereas for $|σ|\gg1$, the measure is not normalisable. This generalises a result of Oh, Okamoto, and Tolomeo (2025) on the $Φ^3_3$-measure. |
| title | Phase transitions for fractional $Φ^3_d$ on the torus |
| topic | Probability Mathematical Physics Analysis of PDEs 60H30 (Primary) 81T08, 82B26, 35Q55 (Secondary) |
| url | https://arxiv.org/abs/2501.17669 |