Deborah Heicklen, Russell Lyons
Call a percolation process on edges of a graph change intolerant if the status of each edge is almost surely determined by the status of the other edges. We give necessary and sufficient conditions for change intolerance of the wired spanning forest when the underlying graph is a spherically symmetric tree.
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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…
Sampling Random Cycle-Rooted Spanning Forests on Infinite Graphs
本文在无限图上构造环根生成森林的无限体积 Gibbs 测度，并提出推广 Wilson 算法的随机游走抽样方法，证明其分量有限且两点相关性指数衰减。
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…
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 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…
We show that for a transitive unimodular graph, the number of ends is the same for every tree of the free minimal spanning forest. This answers a question of Lyons, Peres and Schramm.
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.
The two-dimensional directed spanning forest (DSF) introduced by Baccelli and Bordenave is a planar directed forest whose vertex set is given by a homogeneous Poisson point process $\mathcal{N}$ on $\mathbb{R}^2$. If the DSF has direction $-e_y$, the ancestor $h(u)$ of a vertex $u \in \mathcal{N}$ is the nearest Poisson point (in the $L_2$ distance) having strictly larger $y$-coordinate. This construction induces complex geometrical dependencies. In this paper we show that the collection of DSF paths, properly scaled, converges in distribution to the Brownian web (BW). This verifies a conjectu…
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.
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…
The Euclidean Directed Spanning Forest is a random forest in $\mathbb{R}^d$ introduced by Baccelli and Bordenave in 2007 and we introduce and study here the analogous tree in the hyperbolic space. The topological properties of the Euclidean DSF have been stated for $d=2$ and conjectured for $d \ge 3$ (see further): it should be a tree for $d \in \{2,3\}$ and a countable union of disjoint trees for $d \ge 4$. Moreover, it should not contain bi-infinite branches whatever the dimension $d$. In this paper, we construct the Hyperbolic Directed Spanning Forest (HDSF) and we give a complete descripti…
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 consider the Directed Spanning Forest (DSF) constructed as follows: given a Poisson point process N on the plane, the ancestor of each point is the nearest vertex of N having a strictly larger abscissa. We prove that the DSF is actually a tree. Contrary to other directed forests of the literature, no Markovian process can be introduced to study the paths in our DSF. Our proof is based on a comparison argument between surface and perimeter from percolation theory. We then show that this result still holds when the points of N belonging to an auxiliary Boolean model are removed. Using these r…
$β$-Skewed Maximal Spanning Forests
本文提出$\beta$-偏斜极大生成森林族，插值于自由极大与自由极小生成森林之间，并证明其有线变体小$\beta$极限与极小森林重合当且仅当$p_h=p_u$，且回答了Terlov和Timár的问题。
We show that for finite-range, symmetric random walks on general transient Cayley graphs, the expected occupation time of any given ball of radius $r$ is $O(r^{5/2})$.. We also study the volume-growth property of the wired spanning forests on general Cayley graphs, showing that the expected number of vertices in the component of the identity inside any given ball of radius $r$ is $O(r^{11/2})$.
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…
Local limit of massive spanning forests on the complete graph
本文确定了完全图上大规模生成森林的局部极限，推广了Grimmett关于均匀生成树局部极限的著名定理。
The scaling limit of the root component in the Wired Minimal Spanning Forest of the Poisson Weighted Infinite Tree
本文证明了泊松加权无限树的有线最小生成森林中根连通分量的标度极限，该极限由独立布朗树通过两种粘合过程聚合而成。
The two-dimensional $\ell_{\infty}$ Directed Spanning Forest
本文研究了$\ell_{\infty}$有向生成森林的连通性及路径合并时间的尾分布估计，揭示了其几何特性对论证方法的影响。
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…
For different reversible Markov kernels on finite state spaces, we look for families of probability measures for which the time evolution almost remains in their convex hull. Motivated by signal processing problems and metastability studies we are interested in the case when the size of such families is smaller than the size of the state space, and we want such distributions to be with small overlap among them. To this aim we introduce a squeezing function to measure the common overlap of such families, and we use random forests to build random approximate solutions of the associated intertwin…
The wired minimal spanning forest on the Poisson-weighted infinite tree
本研究针对泊松加权无限树上的有线最小生成森林，通过分析其含根树的简单随机游走，得出谱维数与典型位移指数的具体值，验证了此前的相关预测。
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 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…
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).
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…
We give a lower bound on the expected degree of the free minimal spanning forest of a vertex transitive graph in terms of its spectral radius. This result answers a question of Lyons-Peres-Schramm and simplifies the Gaboriau-Lyons proof of the measurable-group-theoretic solution to von Neumann's problem. In the second part we study a relative version of the free minimal spanning forest. As a consequence of this study we can show that non-torsion unitarizable groups have fixed price one.
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 study random two-component spanning forests ($2$SFs) of finite graphs, giving formulas for the first and second moments of the sizes of the components, vertex-inclusion probabilities for one or two vertices, and the probability that an edge separates the components. We compute the limit of these quantities when the graph tends to an infinite periodic graph in ${\mathbb R}^d$.
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.
Temperley's bijection relates the toroidal dimer model to cycle rooted spanning forests ($CRSF$) on the torus. The height function of the dimer model and the homology class of $CRSF$ are naturally related. When the size of the torus tends to infinity, we show that the measure on $CRSF$ arising from the dimer model converges to a measure on (disconnected) spanning forests or spanning trees. There is a phase transition, which is determined by the average height change.
A zero-one law for swap-invariant events in uniform spanning forests
本文证明均匀生成森林中交换不变事件满足零一律，并应用于分量数与过剩端数的恒定性，解决两个长期未决问题。
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 …
Random walk reflected off of infinity, with applications to uniform spanning forests and supercritical Liouville quantum gravity
本文定义了在瞬态无限图上从无穷远反射的随机游走，证明了Aldous-Broder算法和Wilson算法可生成自由均匀生成森林，并应用于超临界刘维尔量子引力中的随机平面地图。
Spanning trees, cycle-rooted spanning forests on discretizations of flat surfaces and analytic torsion
研究离散化平面表面的图拉普拉斯矩阵行列式渐近展开，关联生成树数量和带循环根生成森林的渐近展开与正则化决定量。
Counting two-forests and random cut size via potential theory
本文利用势论方法，通过顶点数、边数和生成树数给出生成双森林数量的下界，并由此得到随机双森林平均割大小的上界。
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 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…
Perfect Matchings and Essential Spanning Forests in Hyperbolic Double Circle Packings
本文利用双曲图的双圆填充方法，证明了调和 Dirichlet 函数与均匀生成森林极值无限体积测度的存在性，并给出完美匹配柱事件概率的显式公式，进而证明特定边界条件下高度差方差有限，与二维欧氏可容图的 $\log n$ 增长形成对比。
The classical matrix-tree theorem relates the determinant of the combinatorial Laplacian on a graph to the number of spanning trees. We generalize this result to Laplacians on one- and two-dimensional vector bundles, giving a combinatorial interpretation of their determinants in terms of so-called cycle rooted spanning forests (CRSFs). We construct natural measures on CRSFs for which the edges form a determinantal process. This theory gives a natural generalization of the spanning tree process adapted to graphs embedded on surfaces. We give a number of other applications, for example, we compu…
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 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. Negati…
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…
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.
Random Multi-Type Spanning Forests for Synchronization on Sparse Graphs
本研究针对图上的角度同步与平滑问题，提出基于多类型生成森林的扩散估计器，通过沿随机生成子图分支传播实现全局估计，在特定稀疏图实例中优于标准数值线性代数求解器。
Sampling biased spanning forests through marked vertices
研究完全图K_N的κ偏 spanning forest 中连接多个顶点的子树分布及其渐进行为，并在κ=O(√N)处识别相变现象。
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…
We construct forests that span $\mathbb{Z}^d$, $d\geq2$, that are stationary and directed, and whose trees are infinite, but for which the subtrees attached to each vertex are as short as possible. For $d\geq3$, two independent copies of such forests, pointing in opposite directions, can be pruned so as to become disjoint. From this, we construct in $d\geq3$ a stationary, polynomially mixing and uniformly elliptic environment of nearest-neighbor transition probabilities on $\mathbb{Z}^d$, for which the corresponding random walk disobeys a certain zero--one law for directional transience.
Determinantal linear spanning forests
本文提出行列式线性生成森林（DLSF），将其推广到含规范场的图场景，研究其支撑集、配分函数及无限图上的性质，明确有线与自由DLSF在常返图上一致、高维Z^d上不一致的结论。
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).
A Cycle Walk for Sampling Measures on Spanning Forests for Redistricting
提出一种新的马尔可夫链蒙特卡洛方法——循环游走，通过结合局部循环移动和全局人口交换移动，在平衡图划分上高效采样，改善了弱生成树计数权重分布下的收敛性。
Spectrum Estimation through Kirchhoff Random Forests
本文提出通过基尔霍夫随机森林轨迹进行频谱估计，利用蒙特卡洛方法高效计算拉普拉斯特征值，并扩展至对称实矩阵的频谱估计。
Fixed forests in the minimum spanning tree and cubic volume growth
本文针对带均匀边权完全图的最小生成树，研究固定森林出现的概率渐近性质，给出相关函数的递归描述与显式结果，还得到最小生成树中球的期望大小渐近值及指数尾界。
Connectedness of the Free Uniform Spanning Forest as a function of edge weights
研究正则树与有限连通传递图笛卡尔积中，自由均匀生成森林（FSF）连通性随边权变化的相变现象，证明了大权值下连通、小权值下不连通，并给出单边图情形的显式距离分布公式。
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 …
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 some Poissonian deletion of edges or points. We show how to relate these scaling limits to a stationary distribution of a natural coalescent-type Markov process on a state-space of abstract graphs with real-valued edge-weights. This Markov process can be interpreted as a renormalization flow. This provides a model for which one can rigorously implement the formalism proposed by the third author in ord…
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…
Let $G$ be an infinite graph such that each tree in the wired uniform spanning forest on $G$ has one end almost surely. On such graphs $G$, we give a family of continuous, measure preserving, almost one-to-one mappings from the wired spanning forest on $G$ to recurrent sandpiles on $G$, that we call anchored burning bijections. In the special case of $\mathbb{Z}^d$, $d \ge 2$, we show how the anchored bijection, combined with Wilson's stacks of arrows construction, as well as other known results on spanning trees, yields a power law upper bound on the rate of convergence to the sandpile measur…
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.
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