CRITICAL GAUSSIAN MULTIPLICATIVE CHAOS: CONVERGENCE OF THE DERIVATIVE MARTINGALE

• 成果类型: Article
• 署名作者: Duplantier, Bertrand; Rhodes, Remi; Sheffield, Scott; Vargas, Vincent
• 署名单位: Universite Paris Saclay; CEA; Centre National de la Recherche Scientifique (CNRS); Universite PSL; Universite Paris-Dauphine; Centre National de la Recherche Scientifique (CNRS); CNRS - Institute of Chemistry (INC); Massachusetts Institute of Technology (MIT)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/13-AOP890
• 发表日期: 2014
• 页码: 1769-1808
• 关键词: branching random-walk; invariant random measures; critical-behavior; QUANTUM-GRAVITY; fixed-points; matrix model; field-theory; smoothing transformation; nonperturbative solution; random surfaces

摘要

In this paper, we study Gaussian multiplicative chaos in the critical case. We show that the so-called derivative martingale, introduced in the context of branching Brownian motions and branching random walks, converges almost surely (in all dimensions) to a random measure with full support. We also show that the limiting measure has no atom. In connection with the derivative martingale, we write explicit conjectures about the glassy phase of log-correlated Gaussian potentials and the relation with the asymptotic expansion of the maximum of log-correlated Gaussian random variables.