MAXIMUM OF THE GINZBURG-LANDAU FIELDS

• 成果类型: Article
• 署名作者: Belius, David; Wu, Wei
• 署名单位: University of Basel; New York University; NYU Shanghai
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/19-AOP1416
• 发表日期: 2020
• 页码: 2647-2679
• 关键词: entropic repulsion; CONVERGENCE; LAW; perturbations; extremes; models

摘要

We study a two-dimensional massless field in a box with potential V(del phi(.)) and zero boundary condition, where V is any symmetric and uniformly convex function. Naddaf-Spencer (Comm. Math. Phys. 183 (1997) 55-84) and Miller (Comm. Math. Phys. 308 (2011) 591-639) proved that the rescaled macroscopic averages of this field converge to a continuum Gaussian free field. In this paper, we prove that the distribution of local marginal cb(x), for any x in the bulk, has a Gaussian tail. We further characterize the leading order of the maximum and the dimension of high points of this field, thus generalizing the results of Bolthausen-Deuschel-Giacomin (Ann. Probab. 29 (2001) 1670-1692) and Daviaud (Ann. Probab. 34 (2006) 962-986) for the discrete Gaussian free field.