Sharp coupling threshold for the three-dimensional cubic Gibbs measure
Let $\mathbb{T}^3 = (\mathbb{R}/2\pi\mathbb{Z})^3$ with normalised Lebesgue measure $dx$, let $\mu$ be the massive Gaussian free field on $\mathbb{T}^3$, the Gaussian measure with Cameron–Martin space $H^1(\mathbb{T}^3)$, and let $u_N = \pi_N u$, where $\pi_N$ is the Fourier projection onto the frequencies $n \in \mathbb{Z}^3$ with $\max_j |n_j| \le N$. Take the Wick powers ${:}u_N^2{:}$ and ${:}u_N^3{:}$ with respect to the variance of $u_N(x)$ under $\mu$. For $\sigma \ne 0$ and $A > 0$ let $$R_N(u) = -\frac{\sigma}{3} \int_{\mathbb{T}^3} {:}u_N^3{:}\, dx + A \left| \int_{\mathbb{T}^3} {:}u_N^2{:}\, dx \right|^3, \qquad d\rho_N(u) = Z_N^{-1} e^{-R_N(u) - \alpha_N}\, d\mu(u),$$ where $\alpha_N$ is Oh, Okamoto and Tolomeo's diverging renormalisation constant and $Z_N$ normalises. Oh, Okamoto and Tolomeo proved that there are $\sigma_1 \ge \sigma_0 > 0$ such that: for $0 < |\sigma| < \sigma_0$ and $A = A(\sigma)$ sufficiently large, $\rho_N$ converges weakly to a unique limit, the $\Phi^3_3$ measure, which is singular with respect to $\mu$; and for $|\sigma| > \sigma_1$ the $\Phi^3_3$ measure is not normalizable, in the sense that a $\sigma$-finite version of it, obtained from tamed truncated measures, has infinite total mass, while $(\rho_N)$ has no weak limit, even along a subsequence, as measures on $B^{-3/4}_{3,\infty}(\mathbb{T}^3)$.
Problem. Is there $\sigma_* > 0$ such that, for every $0 < |\sigma| < \sigma_*$ and a suitable sufficiently large $A = A(\sigma)$, the measures $\rho_N$ converge weakly to a unique limit, while for every $|\sigma| > \sigma_*$ and every $A > 0$ the $\Phi^3_3$ measure is not normalizable in the sense of Oh, Okamoto and Tolomeo? If so, what is $\sigma_*$, and is the measure normalizable at $|\sigma| = \sigma_*$?
Source: Oh, Okamoto and Tolomeo (2025), Remark 1.2.5 (Remark 1.12 of arXiv:2108.06777v3).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- AP, MP
- Keywords
- AIM Open Applied Problems, focusing Gibbs measures, Gaussian free fields, stochastic quantization
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@book{oh2025,
author = {Oh, Tadahiro and Okamoto, Mamoru and Tolomeo, Leonardo},
title = {Stochastic Quantization of the {$\Phi^3_3$}-Model},
series = {Mem. Eur. Math. Soc.},
volume = {16},
publisher = {EMS Press},
address = {Berlin},
year = {2025},
doi = {10.4171/MEMS/16},
eprint = {2108.06777},
archivePrefix = {arXiv},
}
@misc{aim-sharp-coupling-threshold-for-the-three-dimensional-cubic-gibbs-measure,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Sharp coupling threshold for the three-dimensional cubic Gibbs measure}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/996-sharp-coupling-threshold-for-the-three-dimensional-cubic-gibbs-measure.md},
note = {Entry 996 at commit 4f5b165, 2026-11-04},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01161, "Sharp coupling threshold for the three-dimensional cubic Gibbs measure", clopen.solutions.