### Abstract

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 ℝ^{d}. Tightness of the distribution, as δ → 0, is established for the following two-dimensional examples: the uniformly random spanning tree on δℤ^{2}, the minimal spanning tree on δSℤ^{2} (with random edge lengths), and the Euclidean minimal spanning tree on a Poisson process of points in ℝ^{2} with density δ^{-2}. In each case, sample trees are proven to have the following properties, with probability 1 with respect to any of the limiting measures: (i) there is a single route to infinity (as was known for δ > 0); (ii) the tree branches are given by curves which are regular in the sense of Hölder continuity; (iii) the branches are also rough, in the sense that their Hausdorff dimension exceeds 1; (iv) there is a random dense subset of ℝ^{2}, of dimension strictly between 1 and 2, on the complement of which (and only there) the spanning subtrees are unique with continuous dependence on the endpoints; (v) branching occurs at countably many points in ℝ^{2}; 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.

Original language | English (US) |
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Pages (from-to) | 319-367 |

Number of pages | 49 |

Journal | Random Structures and Algorithms |

Volume | 15 |

Issue number | 3-4 |

DOIs | |

State | Published - 1999 |

### All Science Journal Classification (ASJC) codes

- Software
- Mathematics(all)
- Computer Graphics and Computer-Aided Design
- Applied Mathematics

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## Cite this

*Random Structures and Algorithms*,

*15*(3-4), 319-367. https://doi.org/10.1002/(sici)1098-2418(199910/12)15:3/4<319::aid-rsa8>3.0.co;2-g