Uniform random spanning trees

Volume and heat kernel fluctuations for the three-dimensional uniform spanning tree

Let $\mathcal{U}$ be the uniform spanning tree on $\mathbb{Z}^{3}$. We show the occurrence of log-logarithmic fluctuations around the leading order for the volume of intrinsic balls in $\mathcal{U}$. As an application, we obtain similar fluctuations for the quenched heat kernel of the simple random walk on $\mathcal{U}$.

Spanning tree size in Random Binary Search Trees

This paper deals with the size of the spanning tree of p randomly chosen nodes in a binary search tree. It is shown via generating functions methods, that for fixed p, the (normalized) spanning tree size converges in law to the Normal distribution. The special case p=2 reproves the recent result (obtained by the contraction method by Mahmoud and Neininger [Ann. Appl. Probab. 13 (2003) 253-276]), that the distribution of distances in random binary search trees has a Gaussian limit law. In the proof we use the fact that the spanning tree size is closely related to the number of passes in Multipl…

The number of ends in the uniform spanning tree for recurrent unimodular random graphs

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).

关于网格中最小生成树分布的研究

On the Minimum Spanning Tree Distribution in Grids

本文研究了大网格中最小生成树分布的性质，建立了最可能和最不可能生成树的概率衰减率的界限。

有向最小生成树在完全图上（至少局部地）看起来像均匀生成树

Oriented Minimum spanning tree looks like the Uniform spanning tree on the complete graph (at least locally)

研究完全图中有向最小生成树形图的局部极限，证明其与均匀随机树局部极限相同，与最小生成树不同，揭示引入方向改变最小生成树局部几何结构。

Spanning Tree Bounds for Grid Graphs

Among subgraphs with a fixed number of vertices of the regular square lattice, we prove inequalities that essentially say that those with smaller boundaries have larger numbers of spanning trees and vice-versa. As an application, we relate two commonly used measurements of the compactness of district maps.

The number of spanning clusters of the uniform spanning tree in three dimensions

Let ${\mathcal U}_δ$ be the uniform spanning tree on $δ\mathbb{Z}^{3}$. A spanning cluster of ${\mathcal U}_δ$ is a connected component of the restriction of ${\mathcal U}_δ$ to the unit cube $[0,1]^{3}$ that connects the left face $\{ 0 \} \times [0,1]^{2}$ to the right face $\{ 1 \} \times [0,1]^{2}$. In this note, we will prove that the number of the spanning clusters is tight as $δ\to 0$, which resolves an open question raised by Benjamini (1999).

Pfaffian formulas for spanning tree probabilities

We show that certain topologically defined uniform spanning tree probabilities for graphs embedded in an annulus can be computed as linear combinations of Pfaffians of matrices involving the line-bundle Green's function, where the coefficients count cover-inclusive Dyck tilings of skew Young diagrams.

随机选择生成树

Random choice spanning trees

本文提出选择生成树模型，基于选择随机游走构建，证明其缩放极限为随机聚合树的变体，并扩展了环擦除收敛到广义Rayleigh过程的结果。

Asymptotic Enumeration of Spanning Trees

We give new general formulas for the asymptotics of the number of spanning trees of a large graph. A special case answers a question of McKay (1983) for regular graphs. The general answer involves a quantity for infinite graphs that we call "tree entropy", which we show is a logarithm of a normalized determinant of the graph Laplacian for infinite graphs. Tree entropy is also expressed using random walks. We relate tree entropy to the metric entropy of the uniform spanning forest process on quasi-transitive amenable graphs, extending a result of Burton and Pemantle (1993).

有向生成树与有向图的平稳分布

Oriented spanning trees and stationary distribution of digraphs

本文利用有向图的双团划分，建立了有向生成树数目、平稳分布和Kemeny常数的约化公式，并推广了Knuth和Levine的结果。

The diameter of the uniform spanning tree of dense graphs

We show that the diameter of a uniformly drawn spanning tree of a simple connected graph on $n$ vertices with minimal degree linear in $n$ is typically of order $\sqrt{n}$. A byproduct of our proof, which is of independent interest, is that on such graphs the Cheeger constant and the spectral gap are comparable.

The diameter of uniform spanning trees in high dimensions

We show that the diameter of a uniformly drawn spanning tree of a connected graph on $n$ vertices which satisfies certain high-dimensionality conditions typically grows like $Θ(\sqrt{n})$. In particular this result applies to expanders, finite tori $\mathbb{Z}_m^d$ of dimension $d \geq 5$, the hypercube $\{0,1\}^m$, and small perturbations thereof.

Active spanning trees with bending energy on planar maps and SLE-decorated Liouville quantum gravity for $κ> 8$

We introduce a two-parameter family of probability measures on spanning trees of a planar map. One of the parameters controls the activity of the spanning tree and the other is a measure of its bending energy. When the bending parameter is 1, we recover the active spanning tree model, which is closely related to the critical Fortuin--Kasteleyn model. A random planar map decorated by a spanning tree sampled from our model can be encoded by means of a generalized version of Sheffield's hamburger-cheeseburger bijection. Using this encoding, we prove that for a range of parameter values (including…