We consider three probability measures on subsets of edges of a given finite graph $G$, namely those which govern, respectively, a uniform forest, a uniform spanning tree, and a uniform connected subgraph. A conjecture concerning the negative association of two edges is reviewed for a uniform forest, and a related conjecture is posed for a uniform connected subgraph. The former conjecture is verified numerically for all graphs $G$ having eight or fewer vertices, or having nine vertices and no more than eighteen edges, using a certain computer algorithm which is summarised in this paper. Negative association is known already to be valid for a uniform spanning tree. The three cases of uniform forest, uniform spanning tree, and uniform connected subgraph are special cases of a more general conjecture arising from the random-cluster model of statistical mechanics.
We prove the existence of an automorphism-invariant coupling for the wired and the free uniform spanning forests on Cayley graphs of finitely generated residually amenable groups.
Considering the wired uniform spanning forest on a nonunimodular transitive graph, we show that almost surely each tree of the wired uniform spanning forest is light. More generally we study the tilted volumes for the trees in the wired uniform spanning forest. Regarding the free uniform spanning forest, we consider several families of nonunimodular transitive graphs. We show that the free uniform spanning forest is the same as the wired one on Diestel--Leader graphs. For grandparent graphs, we show that the free uniform spanning forest is connected and has branching number bigger than one. We…
In this work, we expose four bijections each allowing to increase (or decrease) one parameter in either uniform random forests with a fixed number of edges and trees, or quadrangulations with a boundary having a fixed number of faces and a fixed boundary length. In particular, this gives a way to sample a uniform quadrangulation with n + 1 faces from a uniform quadrangulation with n faces or a uniform forest with n+1 edges and p trees from a uniform forest with n edges and p trees.
Starting from any graph on $\{1, \ldots, n\}$, consider the Markov chain where at each time-step a uniformly chosen vertex is disconnected from all of its neighbors and reconnected to another uniformly chosen vertex. This Markov chain has a stationary distribution whose support is the set of non-empty forests on $\{1, \ldots, n\}$. The random forest corresponding to this stationary distribution has interesting connections with the uniform rooted labeled tree and the uniform attachment tree. We fully characterize its degree distribution, the distribution of its number of trees, and the limit di…
It has hitherto been known that in a transitive unimodular graph, each tree in the wired spanning forest has only one end a.s. We dispense with the assumptions of transitivity and unimodularity, replacing them with a much broader condition on the isoperimetric profile that requires just slightly more than uniform transience.
We prove that the free uniform spanning forest of any bounded degree proper plane graph is connected almost surely, answering a question of Benjamini, Lyons, Peres and Schramm. We provide a quantitative form of this result, calculating the critical exponents governing the geometry of the uniform spanning forests of transient proper plane graphs with bounded degrees and codegrees. We find that the same exponents hold universally over this entire class of graphs provided that measurements are made using the hyperbolic geometry of their circle packings rather than their usual combinatorial geomet…
In this note we study the geometry of the component of the origin in the Uniform Spanning Forest of $\mathbb{Z}^d$, as well as in the Uniform Spanning Tree of wired subgraphs of $\mathbb{Z}^d$, when $d \ge 5$. In particular, we study connectivity properties with respect to the Euclidean and the intrinsic distance. We intend to supplement these with further estimates in the future. We are making this preliminary note available, as one of our estimates is used in work of Bhupatiraju, Hanson and Járai on sandpiles.
We prove that every component of the wired uniform spanning forest (WUSF) is one-ended almost surely in every transient reversible random graph, removing the bounded degree hypothesis required by earlier results. We deduce that every component of the WUSF is one-ended almost surely in every supercritical Galton-Watson tree, answering a question of Benjamini, Lyons, Peres and Schramm. Our proof introduces and exploits a family of Markov chains under which the oriented WUSF is stationary, which we call the wired cycle-breaking dynamics.
We prove the following indistinguishability theorem for $k$-tuples of trees in the uniform spanning forest of $\mathbb{Z}^d$: Suppose that $\mathscr{A}$ is a property of a $k$-tuple of components that is stable under finite modifications of the forest. Then either every $k$-tuple of distinct trees has property $\mathscr{A}$ almost surely, or no $k$-tuple of distinct trees has property $\mathscr{A}$ almost surely. This generalizes the indistinguishability theorem of the author and Nachmias (2016), which applied to individual trees. Our results apply more generally to any graph that has the Liou…
We construct a natural discrete random field on $\mathbb{Z}^{d}$, $d\geq 5$ that converges weakly to the bi-Laplacian Gaussian field in the scaling limit. The construction is based on assigning i.i.d. Bernoulli random variables on each component of the uniform spanning forest, thus defines an associated random function. To our knowledge, this is the first natural discrete model (besides the discrete bi-Laplacian Gaussian field) that converges to the bi-Laplacian Gaussian field.
The uniform spanning forest measure ($\mathsf{USF}$) on a locally finite, infinite connected graph $G$ with conductance $c$ is defined as a weak limit of uniform spanning tree measure on finite subgraphs. Depending on the underlying graph and conductances, the corresponding $\mathsf{USF}$ is not necessarily concentrated on the set of spanning trees. Pemantle~\cite{PR1991} showed that on $\mathbb{Z}^d$, equipped with the unit conductance $ c=1$, $\mathsf{USF}$ is concentrated on spanning trees if and only if $d \leq 4$. In this work we study the $\mathsf{USF}$ associated with conductances induc…
We prove that the uniform spanning forests of $\mathbb{Z}^d$ and $\mathbb{Z}^{\ell}$ have qualitatively different connectivity properties whenever $\ell >d \geq 4$. In particular, we consider the graph formed by contracting each tree of the uniform spanning forest down to a single vertex, which we call the component graph. We introduce the notion of ubiquitous subgraphs and show that the set of ubiquitous subgraphs of the component graph changes whenever the dimension changes and is above $8$. To separate dimensions $5,6,7,$ and $8$, we prove a similar result concerning ubiquitous subhypergrap…
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 connected components of the forest from each other by invariantly defined graph properties almost surely. This confirms a conjecture of Benjamini, Lyons, Peres and Schramm. We use this to answer positively two additional questions of Benjamini, Lyons, Peres and Schramm under the assumption of unimodularity. We prove that on any unimodular random rooted network, the FUSF is either connected or has i…
We extend the Aldous-Broder algorithm to generate the wired uniform spanning forests (WUSFs) of infinite, transient graphs. We do this by replacing the simple random walk in the classical algorithm with Sznitman's random interlacement process. We then apply this algorithm to study the WUSF, showing that every component of the WUSF is one-ended almost surely in any graph satisfying a certain weak anchored isoperimetric condition, that the number of `excessive ends' in the WUSF is non-random in any graph, and also that every component of the WUSF is one-ended almost surely in any transient unimo…
In this article we investigate the Uniform Spanning Forest ($\mathsf{USF}$) in the nearest-neighbour integer lattice $\mathbf{Z}^{d+1} = \mathbf{Z}\times \mathbf{Z}^d$ with an assignment of conductances that makes the underlying (Network) Random Walk ($\mathsf{NRW}$) drifted towards the right of the first coordinate. This assignment of conductances has exponential growth and decay; in particular, the measure of balls can be made arbitrarily close to zero or arbitrarily large. We establish upper and lower bounds for its Green's function. We show that in dimension $d = 1, 2$ the $\mathsf{USF}$ c…
Given a subgraph $H$ of a graph $G$, the induced graph of $H$ is the largest subgraph of $G$ whose vertex set is the same as that of $H$. Our paper concerns the induced graphs of the components of $\operatorname{WSF}(G)$, the wired spanning forest on $G$, and, to a lesser extent, $\operatorname{FSF}(G)$, the free uniform spanning forest. We show that the induced graph of each component of $\operatorname{WSF}(\mathbb Z^d$) is almost surely recurrent when $d\ge 8$. Moreover, the effective resistance between two points on the ray of the tree to infinity within a component grows linearly when $d\g…
The uniform spanning forest (USF) in Z^d is the weak limit of random, uniformly chosen, spanning trees in [-n,n]^d. Pemantle proved that the USF consists a.s. of a single tree if and only if d <= 4. We prove that any two components of the USF in Z^d are adjacent a.s. if 5 <= d <= 8, but not if d >= 9. More generally, let N(x,y) be the minimum number of edges outside the USF in a path joining x and y in Z^d. Then a.s. max{N(x,y) : x,y in Z^d} is the integer part of (d-1)/4. The notion of stochastic dimension for random relations in the lattice is introduced and used in the proof.
We study rotor walk, a deterministic counterpart of the simple random walk, on infinite transient graphs. We show that the final rotor configuration of the rotor walk follows the law of the wired uniform spanning forest oriented toward infinity (OWUSF) measure when the initial rotor configuration is sampled from OWUSF. This result holds for all graphs for which each tree in the wired spanning forest has one single end almost surely. This answers a question posed in a previous work of the author (Chan 2018).
We prove the rather counterintuitive result that there exist finite transitive graphs H and integers k such that the Free Uniform Spanning Forest in the direct product of the k-regular tree and H has infinitely many trees almost surely. This shows that the number of trees in the FUSF is not a quasi-isometry invariant. Moreover, we give two different Cayley graphs of the same virtually free group such that the FUSF has infinitely many trees in one, but is connected in the other, answering a question of Lyons and Peres (2016) in the negative. A version of our argument gives an example of a non-u…
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 uniform spanning tree, these measures arise naturally from the laplacian on the graph. More generally these results hold for the "massive" laplacian determinant which counts rooted spanning forests with weight $M$ per finite component. These measures typically have a form of conformal invariance, unlike the usual rooted spanning tree measure. We show that the spectral curve for these models is alwa…
An essential spanning forest of an infinite graph $G$ is a spanning forest of $G$ in which all trees have infinitely many vertices. Let $G_n$ be an increasing sequence of finite connected subgraphs of $G$ for which $\bigcup G_n=G$. Pemantle's arguments imply that the uniform measures on spanning trees of $G_n$ converge weakly to an $\operatorname {Aut}(G)$-invariant measure $μ_G$ on essential spanning forests of $G$. We show that if $G$ is a connected, amenable graph and $Γ\subset \operatorname {Aut}(G)$ acts quasitransitively on $G$, then $μ_G$ is the unique $Γ$-invariant measure on essential…
We study rotor walks on transient graphs with initial rotor configuration sampled from the oriented wired uniform spanning forest (OWUSF) measure. We show that the expected number of visits to any vertex by the rotor walk is at most equal to the expected number of visits by the simple random walk. In particular, this implies that this walk is transient. When these two numbers coincide, we show that the rotor configuration at the end of the process also has the law of OWUSF. Furthermore, if the graph is vertex-transitive, we show that the average number of visits by $n$ consecutive rotor walks …
We prove that the infinite components of the Free Uniform Spanning Forest of a Cayley graph are indistinguishable by any invariant property, given that the forest is different from its wired counterpart. Similar result is obtained for the Free Minimal Spanning Forest. We also show that with the above assumptions there can only be 0, 1 or infinitely many components. These answer questions by Benjamini, Lyons, Peres and Schramm. Our methods apply to a more general class of percolations, those satisfying "weak insertion tolerance", and work beyond Cayley graphs, in the more general setting of uni…
A class of graphs is bridge-addable if given a graph $G$ in the class, any graph obtained by adding an edge between two connected components of $G$ is also in the class. The authors recently proved a conjecture of McDiarmid, Steger, and Welsh stating that if $\mathcal{G}$ is bridge-addable and $G_n$ is a uniform $n$-vertex graph from $\mathcal{G}$, then $G_n$ is connected with probability at least $(1+o_n(1))e^{-1/2}$. The constant $e^{-1/2}$ is best possible since it is reached for the class of all forests. In this paper we prove a form of uniqueness in this statement: if $\mathcal{G}$ is a b…
This paper is a variation on the uniform spanning tree theme. We use random spanning forests to solve the following problem: for a Markov process on a finite set of size $n$, find a probability law on the subsets of any given size $m \leq n$ with the property that the mean hitting time of such a random target does not depend on the starting point of the random walk. We then explore the connection between random spanning forests and infinitesimal generator spectrum. In particular we give an almost probabilistic proof of an algebraic result due to Micchelli and Willoughby and used by Fill and Mi…
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 degree nonamenable graph. Several of our results are new even in the case of $\mathbb{Z}^d$, $d\geq 5$. In particular, we prove that every tree in the forest has spectral dimension $4/3$ and walk dimension $3$ almost surely, and that the critical exponents governing the intrinsic diameter and volume of the past of a vertex in the forest are $1$ and $1/2$ respectively. (The past of a vertex in the unif…
D. Wilson~\cite{[Wi]} in the 1990's described a simple and efficient algorithm based on loop-erased random walks to sample uniform spanning trees and more generally weighted trees or forests spanning a given graph. This algorithm provides a powerful tool in analyzing structures on networks and along this line of thinking, in recent works~\cite{AG1,AG2,ACGM1,ACGM2} we focused on applications of spanning rooted forests on finite graphs. The resulting main conclusions are reviewed in this paper by collecting related theorems, algorithms, heuristics and numerical experiments. A first foundational …
We prove that if a unimodular random rooted graph is recurrent, the number of ends of its uniform spanning tree is almost surely equal to the number of ends of the graph. Together with previous results in the transient case, this completely resolves the problem of the number of ends of wired uniform spanning forest components in unimodular random rooted graphs and confirms a conjecture of Aldous and Lyons (2006).
In this paper, we consider the random plane forest uniformly drawn from all possible plane forests with a given degree sequence. Under suitable conditions on the degree sequences, we consider the limit of a sequence of such forests with the number of vertices tends to infinity in terms of Gromov-Hausdorff-Prokhorov topology. This work falls into the general framework of showing convergence of random combinatorial structures to certain Gromov-Hausdorff scaling limits, described in terms of the Brownian Continuum Random Tree, pioneered by the work of Aldous. In fact we identify the limiting rand…
We introduce a random finite rooted tree $\mathcal{C}$, the steady state cluster, characterized by a recursive description: $\mathcal{C}$ is a singleton with probability $1/2$ and otherwise is obtained by joining by an edge the roots of two independent trees $\mathcal{C}'$ and $\mathcal{C}''$, each having the law of $\mathcal{C}$, then re-rooting the resulting tree at a uniform random vertex. We construct a stationary regenerative stochastic process $\mathcal{C}(t)$, the steady state cluster growth process. It is characterized by a simple fixed-point property. Its stationary distribution is th…
Stochastic networks based on random point sets as nodes have attracted considerable interest in many applications, particularly in communication networks, including wireless sensor networks, peer-to-peer networks and so on. The study of such networks generally requires the nodes to be independently and uniformly distributed as a Poisson point process. In this work, we venture beyond this standard paradigm and investigate the stochastic geometry of networks obtained from \textit{directed spanning forests} (DSF) based on randomly perturbed lattices, which have desirable statistical properties as…
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.
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 (free minimal spanning forest) and WMSF (wired minimal spanning forest), respectively. The WMSF is also the union of the trees that arise from invasion percolation started at all vertices. We show that on any Cayley graph where critical percolation has no infinite clusters, all the component trees in the WMSF have one end a.s. In $\mathbb{Z}^d$ this was proved by Alexander [Ann. Probab. 23 (1995) 87--1…
Tail exponents of the three-dimensional uniform spanning tree and Abelian sandpile
Beijing International Center for Mathematical Research, Peking University（北京大学数学中心）；School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Graduate School of Informatics, Kyoto University（京都大学情报学府）
We study (unrooted) random forests on a graph where the probability of a forest is multiplicatively weighted by a parameter $β>0$ per edge. This is called the arboreal gas model, and the special case when $β=1$ is the uniform forest model. The arboreal gas can equivalently be defined to be Bernoulli bond percolation with parameter $p=β/(1+β)$ conditioned to be acyclic, or as the limit $q\to 0$ with $p=βq$ of the random cluster model. It is known that on the complete graph $K_{N}$ with $β=α/N$ there is a phase transition similar to that of the Erdős--Rényi random graph: a giant tree percolates …
Negative association for a family of random variables $(X_i)$ means that for any coordinate--wise increasing functions $f,g$ we have $$\E f(X_{i_1},...,X_{i_k}) g(X_{j_1},...,X_{j_l}) \leq \E f(X_{i_1},...,X_{i_k}) \E g(X_{j_1},...,X_{j_l})$$ for any disjoint sets of indices $(i_m)$, $(j_n)$. It is a way to indicate the negative correlation in a family of random variables. It was first introduced in 1980s in statistics, and brought to convex geometry in 2005 to prove the Central Limit Theorem for Orlicz balls. The paper gives a relatively simple proof of negative association of absolute values…
