CONVERGENCE IN LAW FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS IN PHASE III

• 成果类型: Article
• 署名作者: Lacoin, Hubert
• 署名单位: Instituto Nacional de Matematica Pura e Aplicada (IMPA)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/21-AOP1551
• 发表日期: 2022
• 页码: 950-983
• 关键词: glassy phase

摘要

Gaussian multiplicative chaos (GMC) is informally defined as a random measure e(gamma X) dx where X is Gaussian field on R-d (or an open subset of it) whose correlation function is of the form K(x, y) = log 1/vertical bar y-x vertical bar + L(x, y), where L is a continuous function of x and y and gamma = alpha + i beta is a complex parameter. In the present paper we consider the case gamma is an element of P-III', where P-III':= {alpha + i beta : alpha, gamma is an element of R, vertical bar alpha vertical bar < root d/2, alpha(2) + beta(2) >= d}. We prove that if X is replaced by an approximation X-epsilon obtained by convolution with a smooth kernel, then the random distribution e(gamma X epsilon) dx, when properly rescaled, has an explicit nontrivial limit in law when epsilon goes to zero. This limit does not depend on the specific convolution kernel which is used to define X-epsilon and can be described as a complex Gaussian white noise with a random intensity given by a real GMC associated with parameter 2 alpha.