The dynamical Ising-Kac model in 3D converges to Φ34

• 成果类型: Article
• 署名作者: Grazieschi, P.; Matetski, K.; Weber, H.
• 署名单位: University of Bath; Michigan State University; University of Munster
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-024-01316-x
• 发表日期: 2025
• 页码: 671-778
• 关键词: glauber evolution; potentials

摘要

We consider the Glauber dynamics of a ferromagnetic Ising-Kac model on a three-dimensional periodic lattice of size (2N + 1)(3), in which the flipping rate of each spin depends on an average field in a large neighborhood of radius gamma(-1 )<< N. We study the random fluctuations of a suitably rescaled coarse-grained spin field as N -> infinity and gamma -> 0; we show that near the mean-field value of the critical temperature, the process converges in distribution to the solution of the dynamical Phi(4)(3) model on a torus. Our result settles a conjecture from Giacomin et al. (1999). The dynamical Phi(4)(3) model is given by a non-linear stochastic partial differential equation (SPDE) which is driven by an additive space-time white noise and which requires renormalisation of the non-linearity. A rigorous notion of solution for this SPDE and its renormalisation is provided by the framework of regularity structures (Hairer in Invent Math 198(2):269-504, 2014. https://doi.org/10.1007/s00222-014-0505-4). As in the two-dimensional case (Mourrat and Weber in Commun Pure Appl Math 70(4):717-812, 2017), the renormalisation corresponds to a small shift of the inverse temperature of the discrete system away from its mean-field value.