A CRITERION FOR CONVERGENCE TO SUPER-BROWNIAN MOTION ON PATH SPACE

• 成果类型: Article
• 署名作者: van der Hofstad, Remco; Holmes, Mark; Perkins, Edwin A.
• 署名单位: Eindhoven University of Technology; University of Auckland; University of British Columbia
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/14-AOP953
• 发表日期: 2017
• 页码: 278-376
• 关键词: upper critical dimension; critical oriented percolation; self-avoiding walks; r-point functions; lace expansion; contact process; lattice trees; survival probability; inductive approach; triangle condition

摘要

We give a sufficient condition for tightness for convergence of rescaled critical spatial structures to the canonical measure of super-Brownian motion. This condition is formulated in terms of the r-point functions for r = 2,..., 5. The r-point functions describe the expected number of particles at given times and spatial locations, and have been investigated in the literature for many high-dimensional statistical physics models, such as oriented percolation and the contact process above 4 dimensions and lattice trees above 8 dimensions. In these settings, convergence of the finite-dimensional distributions is known through an analysis of the r-point functions, but the lack of tightness has been an obstruction to proving convergence on path space. We apply our tightness condition first to critical branching random walk to illustrate the method as tightness here is well known. We then use it to prove tightness for sufficiently spread-out lattice trees above 8 dimensions, thus proving that the measure-valued process describing the distribution of mass as a function of time converges in distribution to the canonical measure of super-Brownian motion. We conjecture that the criterion will also apply to other statistical physics models.