A mating-of-trees approach for graph distances in random planar maps

• 成果类型: Article
• 署名作者: Gwynne, Ewain; Holden, Nina; Sun, Xin
• 署名单位: University of Cambridge; Swiss Federal Institutes of Technology Domain; ETH Zurich; Columbia University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-020-00969-8
• 发表日期: 2020
• 页码: 1043-1102
• 关键词: liouville heat kernel; bipolar orientations; walks; MODEL

摘要

We introduce a general technique for proving estimates for certain random planar maps which belong to the gamma-Liouville quantum gravity (LQG) universality class for gamma is an element of(0,2) The family of random planar maps we consider are those which can be encoded by a two-dimensional random walk with i.i.d. increments via a mating-of-trees bijection, and includes the uniform infinite planar triangulation (UIPT;gamma=8/3and planar maps weighted by the number of different spanning trees (gamma=2 bipolar orientations (gamma=4/3or Schnyder woods (gamma=1that can be put on the map. Using our technique, we prove estimates for graph distances in the above family of random planar maps. In particular, we obtain non-trivial upper and lower bounds for the cardinality of a graph distance ball consistent with the Watabiki (Prog Theor Phys Suppl 114:1-17, 1993) prediction for the Hausdorff dimension of gamma-LQG and we establish the existence of an exponent for certain distances in the map. The basic idea of our approach is to compare a given random planar mapMto amated-CRT map-a random planar map constructed from a correlated two-dimensional Brownian motion-using a strong coupling (Zaitsev in ESAIM Probab Stat 2:41-108, 1998) of the encoding walk forMand the Brownian motion used to construct the mated-CRT map. This allows us to deduce estimates for graph distances inMfrom the estimates for graph distances in the mated-CRT map which we proved (using continuum theory) in a previous work. In the special case when gamma=8/3we instead deduce estimates for the8/3-mated-CRT map from known results for the UIPT. The arguments of this paper do not directly use SLE/LQG, and can be read without any knowledge of these objects.