Scaling limits for minimal and random spanning trees in two dimensions

✍ Scribed by Michael Aizenman; Almut Burchard; Charles M. Newman; David B. Wilson

• Publisher: John Wiley and Sons
• Year: 1999
• Tongue: English
• Weight: 469 KB
• Volume: 15
• Category: Article
• ISSN: 1042-9832
• DOI: 10.1002/(sici)1098-2418(199910/12)15:3/4<319::aid-rsa8>3.0.co;2-g

✦ Synopsis

A general formulation is presented for continuum scaling limits of stochastic spanning trees. A spanning tree is expressed in this limit through a consistent collection of subtrees, which includes a tree for every finite set of endpoints in ‫ޒ‬ d . Tightness of the distribution, as ␦ ª 0, is established for the following two-dimensional examples: the 2 2 Ž uniformly random spanning tree on ␦ ‫ޚ‬ , the minimal spanning tree on ␦ ‫ޚ‬ with random .

2 edge lengths , and the Euclidean minimal spanning tree on a Poisson process of points in ‫ޒ‬ with density ␦ y2 . In each case, sample trees are proven to have the following properties, Ž . with probability 1 with respect to any of the limiting measures: i there is a single route to Ž . Ž . infinity as was known for ␦ ) 0 ; ii the tree branches are given by curves which are regular Ž . in the sense of Holder continuity; iii the branches are also rough, in the sense that their ¨Ž .

2

Hausdorff dimension exceeds 1; iv there is a random dense subset of ‫ޒ‬ , of dimension Ž . strictly between 1 and 2, on the complement of which and only there the spanning subtrees Ž . are unique with continuous dependence on the endpoints; v branching occurs at countably 2 Ž . many points in ‫ޒ‬ ; and vi the branching numbers are uniformly bounded. The results include tightness for the loop-erased random walk in two dimensions. The proofs proceed through the derivation of scale-invariant power bounds on the probabilities of repeated crossings of annuli.