THE DISCRETE GAUSSIAN MODEL, I. RENORMALISATION GROUP FLOW AT HIGH TEMPERATURE

• 成果类型: Article
• 署名作者: Bauerschmidt, Roland; Park, Jiwoon; Rodriguez, Pierre -Francois
• 署名单位: University of Cambridge; Imperial College London
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/23-AOP1658
• 发表日期: 2024
• 页码: 1253-1359
• 关键词: kosterlitz-thouless transition; finite-range decomposition; Scaling Limit; perturbations; fluctuations; component; systems

摘要

The Discrete Gaussian model is the lattice Gaussian free field conditioned to be integer -valued. In two dimensions and at sufficiently high temperature, we show that its macroscopic scaling limit on the torus is a multiple of the Gaussian free field. Our proof starts from a single renormalisation group step after which the integer -valued field becomes a smooth field, which we then analyse using the renormalisation group method. This paper also provides the foundation for the construction of the scaling limit of the infinite -volume gradient Gibbs state of the Discrete Gaussian model in the companion paper. Moreover, we develop all estimates for general finite -range interaction with sharp dependence on the range. We expect these estimates to prepare for a future analysis of the spread -out version of the Discrete Gaussian model at its critical temperature.