DETERMINANTAL SPANNING FORESTS ON PLANAR GRAPHS

• 成果类型: Article
• 署名作者: Kenyon, Richard
• 署名单位: Brown University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/18-AOP1276
• 发表日期: 2019
• 页码: 952-988
• 关键词: erased random-walks; conformal-invariance; dimers; trees; entropy; amebas

摘要

We generalize the uniform spanning tree to construct a family of determinantal measures on essential spanning forests on periodic planar graphs in which every component tree is bi-infinite. Like the uniform spanning tree, these measures arise naturally from the Laplacian on the graph. More generally, these results hold for the massive Laplacian determinant which counts rooted spanning forests with weight M per finite component. These measures typically have a form of conformal invariance, unlike the usual rooted spanning tree measure. We show that the spectral curve for these models is always a simple Harnack curve; this fact controls the decay of edge-edge correlations in these models. We construct a limit shape theory in these settings, where the limit shapes are defined by measured foliations of fixed isotopy type.