Abstract

Let G be a finite connected graph. A spanning tree T of G is a connected subgraph of G that contains no cycles and such that every vertex of G is incident to at least one edge of T. The set of spanning trees of a given finite connected graph is obviously finite and hence we may draw one uniformly at random. This random tree is called the uniform spanning tree (UST) of G. This model was first studied by Kirchhoff who gave a formula for the number of spanning trees of a given graph and provided a beautiful connection with the theory of electric networks. In particular, he showed that the probability that a given edge {x, y} of G is contained in the UST equals \(\mathcal {R}_{\mathrm {eff}}(x \leftrightarrow y; G)\); we prove this fundamental formula in Sect. 7.2 (see Theorem 7.2).

Highlights

  • Let G be a finite connected graph

  • The set of spanning trees of a given finite connected graph is obviously finite and we may draw one uniformly at random. This random tree is called the uniform spanning tree (UST) of G. This model was first studied by Kirchhoff [49] who gave a formula for the number of spanning trees of a given graph and provided a beautiful connection with the theory of electric networks

  • We will define there the wired uniform spanning forest which is obtained by taking a limit of the UST probability measures over exhaustions with wired boundary

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Summary

Introduction

Let G be a finite connected graph. A spanning tree T of G is a connected subgraph of G that contains no cycles and such that every vertex of G is incident to at least one edge of T. The set of spanning trees of a given finite connected graph is obviously finite and we may draw one uniformly at random. This random tree is called the uniform spanning tree (UST) of G This model was first studied by Kirchhoff [49] who gave a formula for the number of spanning trees of a given graph and provided a beautiful connection with the theory of electric networks. He showed that the probability that a given edge {x, y} of G is contained in the UST equals Reff(x ↔ y; G); we prove this fundamental formula in Sect.

Uniform Spanning Trees of Planar Graphs of F satisfies
Basic Properties of the UST formula mentioned earlier and the spatial
Basic Properties of the UST Kirchhoff’s Effective Resistance Formula
Uniform Spanning Trees of Planar Graphs suffices to show that
Limits over Exhaustions
Planar Duality
Connectivity of the Free Forest Last Note on Infinite Networks
Method of Random Sets
Exercises

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