DIMERS AND IMAGINARY GEOMETRY

被引:16
作者
Berestycki, Nathanael [1 ]
Laslier, Benoit [2 ]
Ray, Gourab [3 ]
机构
[1] Univ Wien, Fak Math, Vienna, Austria
[2] Univ Paris Diderot, UFR Math, Paris, France
[3] Univ Victoria, Dept Math & Stat, Victoria, BC, Canada
基金
英国工程与自然科学研究理事会; 加拿大自然科学与工程研究理事会;
关键词
Dimer model; imaginary geometry; uniform spanning tree; SLE; Gaussian free field; ERASED RANDOM-WALKS; CONFORMAL-INVARIANCE; HEIGHT FLUCTUATIONS; QUANTUM-GRAVITY; SLE; TILINGS; LATTICE;
D O I
10.1214/18-AOP1326
中图分类号
O21 [概率论与数理统计]; C8 [统计学];
学科分类号
020208 ; 070103 ; 0714 ;
摘要
We show that the winding of the branches in a uniform spanning tree on a planar graph converge in the limit of fine mesh size to a Gaussian free field. The result holds assuming only convergence of simple random walk to Brownian motion and a Russo-Seymour-Welsh type crossing estimate, thereby establishing a strong form of universality. As an application, we prove universality of the fluctuations of the height function associated to the dimer model, in several situations. The proof relies on a connection to imaginary geometry, where the scaling limit of a uniform spanning tree is viewed as a set of flow lines associated to a Gaussian free field. In particular, we obtain an explicit construction of the a.s. unique Gaussian free field coupled to a continuum uniform spanning tree in this way, which is of independent interest.
引用
收藏
页码:1 / 52
页数:52
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