UNIVERSALITY IN MARGINALLY RELEVANT DISORDERED SYSTEMS

• 成果类型: Article
• 署名作者: Caravenna, Francesco; Sun, Rongfeng; Zygouras, Nikos
• 署名单位: University of Milano-Bicocca; National University of Singapore; University of Warwick
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/17-AAP1276
• 发表日期: 2017
• 页码: 3050-3112
• 关键词: stochastic heat-equation; central-limit-theorem; directed polymer; Invariance; BEHAVIOR; chaos

摘要

We consider disordered systems of a directed polymer type, for which disorder is so-called marginally relevant. These include the usual (short-range) directed polymer model in dimension (2 + 1), the long-range directed polymer model with Cauchy tails in dimension (1 + 1) and the disordered pinning model with tail exponent 1/2. We show that in a suitable weak disorder and continuum limit, the partition functions of these different models converge to a universal limit: a log-normal random field with a multi-scale correlation structure, which undergoes a phase transition as the disorder strength varies. As a by-product, we show that the solution of the two-dimensional stochastic heat equation, suitably regularized, converges to the same limit. The proof, which uses the celebrated fourth moment theorem, reveals an interesting chaos structure shared by all models in the above class.