THE SCALING LIMITS OF THE MINIMAL SPANNING TREE AND INVASION PERCOLATION IN THE PLANE

• 成果类型: Article
• 署名作者: Garban, Christophe; Pete, Gabor; Schramm, Oded
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Ecole Centrale de Lyon; Institut National des Sciences Appliquees de Lyon - INSA Lyon; Universite Claude Bernard Lyon 1; Universite Jean Monnet; CNRS - National Institute for Mathematical Sciences (INSMI); Hungarian Academy of Sciences; HUN-REN; HUN-REN Alfred Renyi Institute of Mathematics; Budapest University of Technology & Economics; Microsoft
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/17-AOP1252
• 发表日期: 2018
• 页码: 3501-3557
• 关键词: near-critical percolation; erased random-walks; 2 dimensions; dynamical percolation; conformal-invariance; infinite cluster; forests; exponents

摘要

We prove that the Minimal Spanning Tree and the Invasion Percolation Tree on a version of the triangular lattice in the complex plane have unique scaling limits, which are invariant under rotations, scalings, and, in the case of the MST, also under translations. However, they are not expected to be conformally invariant. We also prove some geometric properties of the limiting MST. The topology of convergence is the space of spanning trees introduced by Aizenman et al. [Random Structures Algorithms 15 (1999) 319-365], and the proof relies on the existence and conformal covariance of the scaling limit of the near-critical percolation ensemble, established in our earlier works.