Noga Alon, Asaf Nachmias, Matan Shalev
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.
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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.
Diameter of uniform spanning trees on random weighted graphs
本文研究随机加权图上均匀生成树的直径，证明在有界度扩展图和有限盒子中，其直径以高概率为 n^{1/2+o(1)} 阶。
A sharp threshold for minimum bounded-depth and bounded-diameter spanning trees and Steiner trees in random networks
在n阶完全图中，当每条边的权是指数随机变量时，Frieze证明了最小生成树的权趋于zeta（3）=1/1^3+1/2^3+1/3^3+1/2^3 + 1/2当n趋于无穷大时。我们认为生成树的深度限制为从一个指定的根k。证明了当k > log_2 log n+omega（1）时，最小有界深度生成树的权在n ->无穷大时仍趋于zeta（3），当k < log_2 log n时，权在log_2 log n-k中是双指数大的.寻找最小有界深度生成树是NP难的，但当k < log_2 log n - omega（1）时，一个简单的贪婪算法是渐近最优的，而当k > log_2 log n+omega（1）时，一个对最小（无界深度）生成树做微小改变的算法是渐近最优的。我们证明了类似的结果，最小有界深度斯坦纳树，树必须连接一个指定的一组m个顶点，可能会或可能不会包括其他顶点。特别地，当m = const * n时，如果k > log_2 log n+omega（1），则完全图上的最小有界深度Steiner树与最小Steiner树具有渐近相同的权重，并且如果1 <= k <= log_2 log n-omega（1），权重在期望和概率上都趋于（1-2^{-k}）sqrt{8 m/n} [sqrt{2 mn}/2^k]^{1/（2^k-1）}。当直径界为2k时，同样的结果也适用于最小有界直径Stein…
We study a new type of random minimum spanning trees. It is built on the complete graph where each vertex is given a weight, which is a positive real number. Then, each edge is given a capacity which is a random variable that only depends on the product of the weights of its endpoints. We then study the minimum spanning tree corresponding to the edge capacities. Under a condition of finite moments on the node weights, we show that the expected diameter and typical distances of this minimum spanning tree are of order $n^{1/3}$. This is a generalization of the results of Addario-Berry, Broutin, …
Random spanning trees in random environment
本文提出了一种名为随机环境中的随机生成树（RSTRE）的新生成树模型，该模型在逆温度（ disorder strength）β变化时介于均匀生成树和最小生成树之间。在具有n个顶点的完全图上，研究发现：在低 disorder 区域（β≤Cn/logn）中，随机生成树的直径通常为n^{1/2}量级，与均匀生成树相同；在高 disorder 区域（β≥n^{4/3}logn）中，直径通常为n^{1/3}量级，与最小生成树相同。作者推测当β=n^α（α∈(1,4/3)）时，直径为n^{γ+o(1)}量级，其中γ=γ(α)严格介于1/2和1/3之间。
Observables of random spanning trees in random environment
本文研究随机环境中随机生成树（RSTRE）的观测量随无序强度变化，对比均匀生成树与最小生成树，给出直径和局部观测量的结果，并包含欧几里得格及超临界随机图的新结论。
We introduce a Markov Chain Monte Carlo algorithm which samples from the space of spanning trees of complete graphs using local rewiring operations only. The probability distribution of graphs of this kind is shown to depend on the symmetries of these graphs, which are reflected in the equilibrium distribution of the Markov chain. We prove that the algorithm is ergodic and proceed to estimate the probability distribution for small graph ensembles with exactly known probabilities. The autocorrelation time of the graph diameter demonstrates that the algorithm generates independent configurations…
We conjecture that finite graphs with positive Cheeger constant admit a spanning subgraph with positive Cheeger constant and girth proportional to the diameter. We prove this conjecture for regular expander graphs with large expansion. Our proof relies on the Local Lemma.
We consider dense graph sequences that converge to a connected graphon and prove that the GHP scaling limit of their uniform spanning trees is Aldous' Brownian CRT. Furthermore, we are able to extract the precise scaling constant from the limiting graphon. As an example, we can apply this to the scaling limit of the uniform spanning trees of the Erdös-Rényi sequence $(G(n,p))_{n \geq 1}$ for any fixed $p \in (0,1]$, and sequences of dense expanders. A consequence of GHP convergence is that several associated quantities of the spanning trees also converge, such as the height, diameter and law o…
The diameter of random spanning trees interpolating between the UST and the MST of the complete graph
本文引入完全图的加权生成树模型$\mathsf{WST}^{β_n}(K_n)$，研究其在UST与MST间的插值相变，证明不同$β_n$范围下直径的增长阶，提出1<α<4/3时直径指数的猜想。
Let $G$ be a connected graph in which almost all vertices have linear degrees and let $T$ be a uniform spanning tree of $G$. For any fixed rooted tree $F$ of height $r$ we compute the asymptotic density of vertices $v$ for which the $r$-ball around $v$ in $T$ is isomorphic to $F$. We deduce from this that if $\{G_n\}$ is a sequence of such graphs converging to a graphon $W$, then the uniform spanning tree of $G_n$ locally converges to a multi-type branching process defined in terms of $W$. As an application, we prove that in a graph with linear minimum degree, with high probability, the densit…
Random minimum spanning tree and dense graph limits
该研究推广了弗里兹1985年的定理，将完全图的最小生成树总权重收敛性推广到图函数极限，并允许边权从不同分布中抽取。
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.
Consider~\(n\) nodes distributed independently across~\(N\) cities contained with the unit square~\(S\) according to a distribution~\(f.\) Each city is modelled as an~\(r_n \times r_n\) square contained within~\(S\) and~\(MSTC_n\) denotes the length of the minimum spanning tree containing all the~\(n\) nodes. We use approximation methods to obtain variance estimates for~\(MSTC_n\) and prove that if the cities are well-connected and densely populated in a certain sense, then~\(MSTC_n\) appropriately centred and scaled converges to zero in probability. Using the proof techniques, we alternately …
On the Color Discrepancy of Spanning Trees in Random and Randomly Perturbed Graphs
本文研究随机图和随机扰动稠密图中生成树的颜色偏差，证明在连通阈值以上的随机图及添加少量随机边的稠密图中，任意2-边着色下都存在具有较大颜色偏差的生成树。
Partition functions for dense instances of combinatorial enumeration problems
给定一个完全图，其边的权重为正，我们定义一个子集的边的权重为该子集中的边的权重的乘积，并考虑各种子集上的权重的和（分拆函数）：圈覆盖，闭行走，生成树。我们表明，如果图的边的权重是在一个恒定的因素，事先固定，彼此然后大部分的分拆函数是集中在一个特别简单的结构的子集：循环覆盖与几个周期，行走，访问每个顶点只有几次，和生成树与每个顶点的小度。这使我们能够构建一个多项式时间算法，以分离图与许多哈密顿圈的图是足够远离哈密尔顿。
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 $M$ 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…
On the diameter of random uniform hypergraphs in dense regime
本文研究了密集区域中随机均匀超图的直径，证明了其直径在两个点上集中，并利用 Stein-Chen 方法和耦合技术得出了相关结论。
Repeat times and a two-weight UST model
本文研究了双权重UST模型中重复次数对生成树直径的影响，通过分析临界图的重复时间估计和直径和的集中界，证明了直径的渐进行为。
Critical Percolation and the Minimal Spanning Tree in Slabs
已知$\mathbb{Z}^{d}$上的最小生成森林对于$d \leq 2$由一棵树组成，而对于大的$d$则被证明是由无穷多棵树组成.本文证明了如$\mathbb{Z}^{2}\times {\{0,\ldots,k\}}^{d-2}$的拟可平面图存在单树。我们的方法依赖于arXiv：1401.7130的“胶合引理”的推广。一个相关的结果是，临界伯努利渗流板满足盒交叉性质。它的证明是基于一个新的准可平面图的Russo-Seymour-Welsh型定理。因此，在临界状态下，从直径为$n$的$0$开始的开放路径的概率在$n$中多项式衰减。这加强了arXiv：1401.7130的结果，其中首次建立了临界状态下不存在无限簇。
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…
Semi-infinite paths of the 2d-Radial Spanning Tree
研究了平面上Poisson点过程的径向生成树的半无限路径。我们首先证明了半无限路径与半径为$r$的球面之间的交点数的期望随$r$的增长呈次线性增长。然后，我们证明，在每个（确定性）的方向，有一个唯一的半无限路径的概率，由无限多的其他半无限路径的密切渐近方向。在$[0,2π)$中，存在多于一条半无限路径的（随机）方向的集合是稠密的。它对应于可能的竞争界面的渐近方向。我们发现，该树可以分解成最多5个无限的子树直接连接到根。分离这些子树的接口进行了研究和模拟。
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