Abstract:We derive the focusing $\Phi^6_1$ measure on the torus $\mathbb{T}$ as the high-temperature/mean-field limit of many-body quantum Gibbs states with an attractive three-body interaction. The main difficulty in the focusing setting is to relate the classical mass cutoff to the quantum particle-number cutoff. Our result reaches the optimal mass threshold for the classical field identified by Oh, Sosoe, and Tolomeo (2022), and thereby extends the earlier work of Rout and Sohinger (2025). At the critical threshold, the short-range interaction is allowed to shrink to a Dirac delta function on a logarithmic scale in the temperature parameter. Strictly below the threshold, the same convergence holds with a polynomial dependence on the temperature. Moreover, we establish a quantum-level phase transition at the same mass threshold. The proof develops the variational framework of Lewin, Nam, and Rougerie (2015) in the focusing setting and relies on two new ingredients: a non-factorized trial state construction and a delicate tail estimate for the interacting lower symbol. These allow us to control the localization and relative entropy errors caused by the particle-number cutoff, as well as the contribution of the focusing exponential weight.
| Comments: | 58 pages |
| Subjects: | Mathematical Physics (math-ph); Analysis of PDEs (math.AP); Probability (math.PR) |
| Cite as: | arXiv:2605.25755 [math-ph] |
| (or arXiv:2605.25755v1 [math-ph] for this version) | |
| https://doi.org/10.48550/arXiv.2605.25755 arXiv-issued DOI via DataCite (pending registration) |
From: Lin Lü [view email]
[v1]
Mon, 25 May 2026 12:07:10 UTC (64 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。