Let x and y be points chosen uniformly at random from $\Z_n^4$, the four-dimensional discrete torus with side length n. We show that the length of the loop-erased random walk from x to y is of order $n^2 (\log n)^{1/6}$, resolving a conjecture of Benjamini and Kozma. We also show that the scaling limit of the uniform spanning tree on $\Z_n^4$ is the Brownian continuum random tree of Aldous. Our proofs use the techniques developed by Peres and Revelle, who studied the scaling limits of the uniform spanning tree on a large class of finite graphs that includes the d-dimensional discrete torus for $d \geq 5$, in combination with results of Lawler concerning intersections of four-dimensional random walks.
We consider uniform spanning tree (UST) in topological polygons with $2N$ marked points on the boundary with alternating boundary conditions. In [LPW21], the authors derive the scaling limit of the Peano curve in the UST. They are variants of SLE$_8$. In this article, we derive the scaling limit of the loop-erased random walk branch (LERW) in the UST. They are variants of SLE$_2$. The conclusion is a generalization of [HLW20,Theorem 1.6] where the authors derive the scaling limit of the LERW branch of UST when $N=2$. When $N=2$, the limiting law is SLE$_2(-1,-1; -1, -1)$. However, the limiting…
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 exponential time of parameter $q>0$. The related random blocks tend to cluster nodes visited by the random walk on time scale $1/q$. This random partitioning is induced by a measure of rooted spanning forest of the graph, which generalizes the classical uniform spanning tree measure and which can be obtained as a zero-limit of FK-percolation with an external cemetery state. Some general properties of this rooted forest measure and related determin…
Convergence of loop-erased random walk in the natural parametrization
环擦除随机游动（Loop-erased Random Walk，简称LERW）是研究最多的临界晶格模型之一。它是一种自避免的随机游动，人们通过考虑均匀选择的生成树中的分支，按顺序或交替地从简单的随机游动中擦除环后得到。本文证明了以重整化长度为参数的平面LERW在5/4维Minkowski容度为参数的SLE（2）的格点尺度极限内收敛。在这样做的时候，我们也提供了一个方法来证明类似的收敛结果收敛到SLE的其他模型。除了主要定理，我们的几个结果关于LERW是独立的利益：例如，两点估计，估计的最大内容，和一个“分离引理”。
D. Wilson~ \cite{[Wi]}在20世纪90年代描述了一种简单而有效的算法，该算法基于环擦除随机游动来采样均匀生成树和更一般地加权的生成给定图的树或森林。该算法提供了一个强大的工具，在分析网络上的结构，并沿着这条思路，在最近的作品~ \cite{AG1,AG2,ACGM1,ACGM2}，我们集中在有限图上的应用程序的生成有根森林。本文通过收集相关的定理、算法、算法和数值实验，对所得到的主要结论进行了综述。第一个基础部分是关于行列式结构和有效的采样过程，接下来是四个主要的应用：1）图中分布良好的点的基于随机行走的概念2）如何通过马尔可夫交织对偶来描述有限设置中的亚稳态动力学3）网络和相关过程的粗粒度方案4）图形信号的小波金字塔算法。
The Aldous--Broder algorithm provides a way of sampling a uniformly random spanning tree for finite connected graphs using simple random walk. Namely, start a simple random walk on a connected graph and stop at the cover time. The tree formed by all the first-entrance edges has the law of a uniform spanning tree. Here we show that the tree formed by all the last-exit edges also has the law of a uniform spanning tree. This answers a question of Tom Hayes and Cris Moore from 2010. The proof relies on a bijection that is related to the BEST theorem in graph theory. We also give other applications…
We show that the law of the three-dimensional uniform spanning tree (UST) is tight under rescaling in a space whose elements are measured, rooted real trees, continuously embedded into Euclidean space. We also establish that the relevant laws actually converge along a particular scaling sequence. The techniques that we use to establish these results are further applied to obtain various properties of the intrinsic metric and measure of any limiting space, including showing that the Hausdorff dimension of such is given by $3/β$, where $β\approx 1.624\dots$ is the growth exponent of three-dimens…
Let $M$ be the infinite spanning-tree-weighted random planar map, which is the local limit of finite random planar maps sampled with probability proportional to the number of spanning trees they admit. We show that a.s. the $M$-graph-distance diameter of the external diffusion-limited aggregation (DLA) cluster on $M$ run for $m$ steps is of order $m^{2/d + o_m(1)}$, where $d$ is the metric ball volume growth exponent for $M$ (which was shown to exist by Ding-Gwynne, 2018). By known bounds for $d$, one has $0.55051\ldots \leq 2/d \leq 0.563315\ldots$. Along the way, we also prove that loop-eras…
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 show that the Brownian continuum random tree is the Gromov-Hausdorff-Prohorov scaling limit of the uniform spanning tree on high-dimensional graphs including the $d$-dimensional torus $\mathbb{Z}_n^d$ with $d>4$, the hypercube $\{0,1\}^n$, and transitive expander graphs. Several corollaries for associated quantities are then deduced: convergence in distribution of the rescaled diameter, height and simple random walk on these uniform spanning trees to their continuum analogues on the continuum random tree.
Guangdong Technion – Israel Institute of Technology（广东以色列理工学院）；Technion – Israel Institute of Technology（以色列理工学院）；Saint Petersburg State University（圣彼得堡国立大学）
