Choosing a Spanning Tree for the Integer Lattice Uniformly
概率论
2007-05-23 v1
摘要
Consider the nearest neighbor graph for the integer lattice Z^d in d dimensions. For a large finite piece of it, consider choosing a spanning tree for that piece uniformly among all possible subgraphs that are spanning trees. As the piece gets larger, this approaches a limiting measure on the set of spanning graphs for Z^d. This is shown to be a tree if and only if d=<4. In this case, the tree has only one topological end, i.e. there are no doubly infinite paths. When d>=5 the spanning forest has infinitely many components almost surely, with each component having one or two topological ends.
引用
@article{arxiv.math/0404043,
title = {Choosing a Spanning Tree for the Integer Lattice Uniformly},
author = {Robin Pemantle},
journal= {arXiv preprint arXiv:math/0404043},
year = {2007}
}
备注
24 pages