# Uniform spanning forests

@article{Benjamini2001UniformSF, title={Uniform spanning forests}, author={Itai Benjamini and Russell Lyons and Yuval Peres and Oded Schramm}, journal={Annals of Probability}, year={2001}, volume={29}, pages={1-65} }

We study uniform spanning forest measures on infinite graphs, which are weak limits of uniform spanning tree measures from finite subgraphs. These limits can be taken with free (FSF) or wired (WSF) boundary conditions. Pemantle proved that the free and wired spanning forests coincide in Z d and that they give a single tree iff d ≤ 4. In the present work, we extend Pemantle's alternative to general graphs and exhibit further connections of uniform spanning forests to random walks, potential…

## 215 Citations

Minimal spanning forests

- Mathematics
- 2004

Minimal spanning forests on infinite graphs are weak limits of minimal spanning trees from finite subgraphs. These limits can be taken with free or wired boundary conditions and are denoted FMSF…

Uniform spanning forests on biased Euclidean lattices

- MathematicsAnnales de l'Institut Henri Poincaré, Probabilités et Statistiques
- 2021

The uniform spanning forest measure (USF) on a locally finite, infinite connected graph with conductance c is defined as a weak limit of uniform spanning tree measure on finite subgraphs. Depending…

Determinantal spanning forests on planar graphs

- MathematicsThe Annals of Probability
- 2019

We generalize the uniform spanning tree to construct a family of determinantal measures on essential spanning forests on periodic planar graphs in which every component tree is bi-infinite. Like the…

Uniqueness of maximal entropy measure on essential spanning forests

- Mathematics
- 2006

An essential spanning forest of an infinite graph G is a spanning forest of G in which all trees have infinitely many vertices. Let Gn be an increasing sequence of finite connected subgraphs of G for…

Near-critical spanning forests and renormalization

- Mathematics
- 2015

We study random two-dimensional spanning forests in the plane that can be viewed both in the discrete case and in their appropriately taken scaling limits as a uniformly chosen spanning tree with…

Universality of high-dimensional spanning forests and sandpiles

- MathematicsProbability Theory and Related Fields
- 2019

We prove that the wired uniform spanning forest exhibits mean-field behaviour on a very large class of graphs, including every transitive graph of at least quintic volume growth and every bounded…

Indistinguishability of trees in uniform spanning forests

- Mathematics
- 2015

We prove that in both the free and the wired uniform spanning forest (FUSF and WUSF) of any unimodular random rooted network (in particular, of any Cayley graph), it is impossible to distinguish the…

Scaling limits for minimal and random spanning trees in two dimensions

- MathematicsRandom Struct. Algorithms
- 1999

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…

Critical Percolation and the Minimal Spanning Tree in Slabs

- Mathematics
- 2015

The minimal spanning forest on ℤd is known to consist of a single tree for d ≤ 2 and is conjectured to consist of infinitely many trees for large d. In this paper, we prove that there is a single…

Loop-erased partitioning of cycle-free graphs

- Mathematics
- 2021

We consider random partitions of the vertex set of a given finite graph that can be sampled by means of loop-erased random walks stopped at a random independent exponential time of parameter q > 0.…

## References

SHOWING 1-10 OF 108 REFERENCES

Uniform and minimal essential spanning forests on trees

- MathematicsRandom Struct. Algorithms
- 1998

The uniform essential spanning forest admits a simple description in terms of a Galton]Watson process and also arises as a limit of random-cluster measures.

Percolation and minimal spanning forests in infinite graphs

- Mathematics
- 1995

The structure of a spanning forest that generalizes the minimal spanning tree is considered for infinite graphs with a value f(b) attached to each bond b. Of particular interest are stationary random…

Scaling limits for minimal and random spanning trees in two dimensions

- MathematicsRandom Struct. Algorithms
- 1999

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…

Scaling limits of loop-erased random walks and uniform spanning trees

- Mathematics
- 1999

AbstractThe uniform spanning tree (UST) and the loop-erased random walk (LERW) are strongly related probabilistic processes. We consider the limits of these models on a fine grid in the plane, as the…

Essential Spanning Forests and Electrical Networks on Groups

- Mathematics
- 1999

AbstractLet Γ be a Cayley graph of a finitely generated group G. Subgraphs which contain all vertices of Γ, have no cycles, and no finite connected components are called essential spanning forests.…

Group-invariant Percolation on Graphs

- Mathematics
- 1999

Abstract. Let G be a closed group of automorphisms of a graph X. We relate geometric properties of G and X, such as amenability and unimodularity, to properties of G-invariant percolation processes…

Critical Random Walk in Random Environment on Trees

- Mathematics
- 1995

We study the behavior of random walk in random environment (RWRE) on trees in the critical case left open in previous work. Representing the random walk by an electrical network, we assume that the…

Generating random spanning trees

- Computer Science, Mathematics30th Annual Symposium on Foundations of Computer Science
- 1989

It is shown that the Markov chain on the space of all spanning trees of a given graph where the basic step is an edge swap is rapidly mixing.

Percolation Perturbations in Potential Theory and Random Walks

- Mathematics
- 1998

We show that on a Cayley graph of a nonamenable group, almost surely the infinite clusters of Bernoulli percolation are transient for simple random walk, that simple random walk on these clusters has…

Infinite clusters in dependent automorphism invariant percolation on trees

- Mathematics
- 1997

We study dependent bond percolation on the homogeneous tree T n of order n ≥ 2 under the assumption of automorphism invariance. Excluding a trivial case, we find that the number of infinite clusters…