The component graph of the uniform spanning forest: transitions in dimensions 9 , 10 , 11 , …


Type
Article
Change log
Authors
Peres, Y 
Abstract

We prove that the uniform spanning forests of Zd and Z have qualitatively different connectivity properties whenever >d≥4. In particular, we consider the graph formed by contracting each tree of the uniform spanning forest down to a single vertex, which we call the component graph. We introduce the notion of ubiquitous subgraphs and show that the set of ubiquitous subgraphs of the component graph changes whenever the dimension changes and is above 8. To separate dimensions 5,6,7, and 8, we prove a similar result concerning ubiquitous subhypergraphs in the component hypergraph. Our result sharpens a theorem of Benjamini, Kesten, Peres, and Schramm, who proved that the diameter of the component graph increases by one every time the dimension increases by four.

Description
Keywords
math.PR, math.PR, math-ph, math.MP
Journal Title
Probability Theory and Related Fields
Conference Name
Journal ISSN
0178-8051
1432-2064
Volume Title
175
Publisher
Springer Science and Business Media LLC
Sponsorship
Microsoft Research