DIMERS AND IMAGINARY GEOMETRY

• 成果类型: Article
• 署名作者: Berestycki, Nathanael; Laslier, Benoit; Ray, Gourab
• 署名单位: University of Vienna; Universite Paris Cite; University of Victoria
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/18-AOP1326
• 发表日期: 2020
• 页码: 1-52
• 关键词: erased random-walks; conformal-invariance; height fluctuations; QUANTUM-GRAVITY; sle; tilings; lattice

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.