Radial Φ4 models in supercritical regimes
Abstract
We consider the Φ4d measure under radial symmetry on the unit ball, formally given by dρ(u) = 1/Z exp( − 1/4 ∫B1 |u|4 dx − 1/2 ∫B1 |∇ u|2 dx) d u, where u is restricted to radially symmetric functions with 0 boundary conditions. We prove that for every dimension d, there exists such a measure, which is concentrated on Lp(B1) for every p < 2d/(d−2) (for every p < ∞ if d ≤ 2). Furthermore, we show that these measures are not Gaussian, and when d ≥ 4, they are singular with respect to the underlying Gaussian measure. We achieve this result by first defining the measure ρ+δ[λ] on the shell {δ≤ |x| ≤ 1}, 0 < δ< 1, with a boundary condition on the sphere of radius δ of the form u|{|x| = δ} = λ. Then we prove that this measure satisfies a variational formula, and use it to show moment bounds which are uniform in the boundary value λ. A crucial ingredient in this argument is the fact that every solution of the elliptic equation −Δu + u3 = 0 on the annulus { δ≤ |x| ≤ 1 } satisfies uniform bounds inside the domain, independently of the boundary conditions.
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BibTeX
@misc{oh2026radial,
title = {{Radial $Φ^4$ models in supercritical regimes}},
author = {Tadahiro Oh and Leonardo Tolomeo and Nikolay Tzvetkov and Yuzhao Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31267},
eprint = {2610.31267},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31267},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Radial Φ^4 models in supercritical regimes
AU - Tadahiro Oh
AU - Leonardo Tolomeo
AU - Nikolay Tzvetkov
AU - Yuzhao Wang
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31267
UR - https://arxiv.org/abs/2610.31267
AB - We consider the Φ^4_d measure under radial symmetry on the unit ball, formally given by dρ(u) = 1/Z exp( − 1/4 ∫_B_1 |u|^4 dx − 1/2 ∫_B_1 |∇ u|^2 dx) d u, where u is restricted to radially symmetric functions with 0 boundary conditions. We prove that for every dimension d, there exists such a measure, which is concentrated on L^p(B_1) for every p < 2d/(d−2) (for every p < ∞ if d ≤ 2). Furthermore, we show that these measures are not Gaussian, and when d ≥ 4, they are singular with respect to the underlying Gaussian measure. We achieve this result by first defining the measure ρ^+_δ[λ] on the shell {δ≤ |x| ≤ 1}, 0 < δ< 1, with a boundary condition on the sphere of radius δ of the form u|_{|x| = δ} = λ. Then we prove that this measure satisfies a variational formula, and use it to show moment bounds which are uniform in the boundary value λ. A crucial ingredient in this argument is the fact that every solution of the elliptic equation −Δu + u^3 = 0 on the annulus { δ≤ |x| ≤ 1 } satisfies uniform bounds inside the domain, independently of the boundary conditions.
ER -