DISTRIBUTION OF LEVELS IN HIGH-DIMENSIONAL RANDOM LANDSCAPES

• 成果类型: Article
• 署名作者: Kabluchko, Zakhar
• 署名单位: Ulm University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/11-AAP772
• 发表日期: 2012
• 页码: 337-362
• 关键词: CENTRAL LIMIT-THEOREMS; local rem conjecture; Empirical Processes; WEAK-CONVERGENCE; energy scales; SEQUENCES; statistics; PROOF

摘要

We prove empirical central limit theorems for the distribution of levels of various random fields defined on high-dimensional discrete structures as the dimension of the structure goes to infinity. The random fields considered include costs of assignments. weights of Hamiltonian cycles and spanning trees, energies of directed polymers, locations of particles in the branching random walk, as well as energies in the Sherrington-Kirkpatrick and Edwards-Anderson models. The distribution of levels in all models listed above is shown to be essentially the same as in a stationary Gaussian process with regularly varying nonsummable covariance function. This type of behavior is different from the Brownian bridge-type limit known for independent or stationary weakly dependent sequences of random variables.