The diameter of a random Cayley graph of Z_q
Gideon Amir, Ori Gurel-Gurevich
考虑具有k个一致选择生成元的素数阶q的循环群的Cayley图。对于固定的k，我们证明了所述图的直径是渐进的（在q）的阶为q^（1/k）。当生成集被取为大小为2k的对称集时，这同样成立。
Consider the Cayley graph of the cyclic group of prime order q with k uniformly chosen generators. For fixed k, we prove that the diameter of said graph is asymptotically (in q) of order q^(1/k). The same also holds when the generating set is taken to be a symmetric set of size 2k.
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We prove the existence of a limiting distribution for the appropriately rescaled diameters of random undirected Cayley graphs of finite nilpotent groups of bounded rank and nilpotency class, thus extending a result of Shapira and Zuck which dealt with the case of abelian groups. The limiting distribution is defined on a space of unimodular lattices, as in the case of random Cayley graphs of abelian groups. Our result, when specialised to a certain family of unitriangular groups, establishes a very recent conjecture of Hermon and Thomas. We derive this as a consequence of a general inequality, …
The thresholds for diameter 2 in random Cayley graphs
给定一个群G，模型$\mathcal{G}(G,p)$表示G的所有Cayley图的概率空间，其中生成集的每个元素以概率p独立地随机选择。在这篇文章中，我们证明了对于任何$ε> 0$和任何n_k阶群G_k族，其中$n_k \to \infty$，如果$p \geqslant \sqrt{(2 + ε) \frac{\log{n_k}}{n_k}}$，则具有高概率的图$Γ_k \in \mathcal{G}(G_k,p)$具有至多2的直径，并且如果$p \leqslant \sqrt{(1/4 + ε)\frac{\log{n_k}}{n_k}}$，则具有高概率的图$Γ_k \in \mathcal{G}(G_k,p)$具有大于2的直径。我们还提供了家庭的图形，这表明这两个结果是最好的可能的例子。特别有趣的是，对于某些群族，相应的随机凯莱图达到直径2的速度明显快于Erdens-Renyi随机图。
We study the diameter of LPS Ramanujan graphs $X_{p,q}$. We show that the diameter of the bipartite Ramanujan graphs is greater than $ (4/3)\log_{p}(n) +O(1)$ where $n$ is the number of vertices of $X_{p,q}$. We also construct an infinite family of $(p+1)$-regular LPS Ramanujan graphs $X_{p,m}$ such that the diameter of these graphs is greater than or equal to $ \lfloor (4/3)\log_{p}(n) \rfloor$. On the other hand, for any $k$-regular Ramanujan graph we show that the distance of only a tiny fraction of all pairs of vertices is greater than $(1+ε)\log_{k-1}(n)$. We also have some numerical expe…
We study the girth of Cayley graphs of finite classical groups G on random sets of generators. Our main tool is an essentially best possible bound we obtain on the probability that a given word w takes the value 1 when evaluated in G in terms of the length of w, which has additional applications. We also study the girth of random directed Cayley graphs of symmetric groups, and the relation between the girth and the diameter of random Cayley graphs of finite simple groups.
Exact Diameter Windows for Random Cayley Graphs on Odd-Order Abelian Groups
本文确定了奇阶阿贝尔群上随机Cayley图的直径临界窗口，证明覆盖时间收敛到泊松过程，直径转变呈Gumbel分布，并给出精确阈值常数\\(d!/2^d\\)及\\(d=2\\)时的路径-环分解。
Random induced subgraphs of Cayley graphs induced by transpositions
本文研究了由任意最小转置生成集诱导的对称群的Cayley图的随机诱导子图。通过选择具有独立概率的置换$λ_n$，得到了这个Cayley图的随机诱导子图.我们的主要结果是：对于任意的换位的最小生成集，对于概率$λ_n=\frac{1+ε_n}{n-1}$，其中$n^{-{1/3}+δ}\le ε_n<1$和$δ>0$，随机诱导子图有a. s.大小为$\wp(ε_n)\frac{1+ε_n}{n-1}n!$的唯一最大组分，其中$\wp(ε_n)$是特定分支过程的存活概率。
Using methods of Marklof and Strömbergsson we establish several limit laws for metric parameters of random Cayley graphs of finite abelian groups with respect to a randomly chosen set of generators of a fixed size. Doing so we settle a conjecture of Amir and Gurel-Gurevich.
We prove that the trace of a transient branching random walk on a planar hyperbolic Cayley graph has a.s. continuum many ends and no isolated end.
Invariant random matchings in Cayley graphs
我们证明了任何不顺从的Cayley图都存在IID完美匹配因子。我们还证明了任何连通的d-正则顶点传递图都允许完美匹配。这两个结果意味着每个Cayley图都有不变的随机完美匹配。证明中的一个关键步骤是关于图形的结果，该结果也适用于有限图形。有限版本说，对于有限正则图的任何部分匹配，这是一个很好的扩展器，总是可以找到一个增广路径的长度是在一个不匹配的顶点的比率的多对数。
Coloring sparse random Cayley graphs
证明了任何有限阿贝尔群上由c log|Z|个随机元素生成的Cayley图以高概率是3-可着色的，改进了Alon的1/4 log log|Z|界，并朝Alon猜想迈进一步。
We consider the random walk in an independent and identically distributed (i.i.d.) random environment on a Cayley graph of a finite free product of copies of $\mathbb{Z}$ and $\mathbb{Z}_2$. Such a Cayley graph is readily seen to be a regular tree. Under a uniform elipticity assumption on the i.i.d. environment we show that the walk has positive speed and establish the annealed central limit theorem for the graph distance of the walker from the starting point.
We present a very simple bijective proof of Cayley's formula due to Foata and Fuchs (1970). This bijection turns out to be very useful when seen through a probabilistic lens; we explain some of the ways in which it can be used to derive probabilistic identities, bounds, and growth procedures for random trees with given degrees, including random d-ary trees. We also introduce a partial order on the degree sequences of rooted trees, and conjecture that it induces a stochastic partial order on heights of random rooted trees with given degrees.
We show that doubling at some large scale in a Cayley graph implies uniform doubling at all subsequent scales. The proof is based on the structure theorem for approximate subgroups proved by Green, Tao and the first author. We also give a number of applications to the geometry and spectrum of finite Cayley graphs. For example, we show that a finite group has moderate growth in the sense of Diaconis and Saloff-Coste if and only if its diameter is larger than a fixed power of the cardinality of the group. We call such groups almost flat and show that they have a subgroup of bounded index admitti…
On the girth of random Cayley graphs
证明了对称群的随机d-正则Cayley图的围长渐近几乎必然至少为（log_{d-1} |G|）^{1/2}/2，F_q上单代数群的随机d-正则Cayley图的围长渐近几乎必然至少为log_{d-1} |G| /dim（G）.对于对称p-群，围长在log log |G|和（log |G|）^alpha之间，其中alpha<1。提出了几个问题和开放的问题。
Random Walks in I.I.D. Random Environment on Cayley Trees
考虑了$d \geq 3$的无限$d$ -正则树上的\emph{i.i.d.}随机环境中的随机游动.我们把树看作是$\Zbold$和$\Zbold_2$的两个拷贝的自由积的Cayley图，并定义了i.i.d.在这个群体的作用下，环境是不变的。在一个温和的非退化假设下，我们证明了行走总是瞬态的。
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 $O(r^{11/2})$.
We prove a quantitative, finitary version of Trofimov's result that a connected, locally finite vertex-transitive graph G of polynomial growth admits a quotient with finite fibres on which the action of Aut(G) is virtually nilpotent with finite vertex stabilisers. We also present some applications. We show that a finite, connected vertex-transitive graph G of large diameter admits a quotient with fibres of small diameter on which the action of Aut(G) is virtually abelian with vertex stabilisers of bounded size. We also show that G has moderate growth in the sense of Diaconis and Saloff-Coste, …
Consider the random Cayley graph of a finite group $G$ with respect to $k$ generators chosen uniformly at random, with $1 \ll k \lesssim \log |G|$. The results of this article supplement those in the three main papers on random Cayley graphs. The majority of the results are inspired by a `universality' conjecture of Aldous and Diaconis (1985). To start, we study the limit profile of cutoff for the simple random walk on this random graph, as well as a detailed investigation into mixing properties when $G = \mathbb Z_p^d$ with $p$ prime. We then exposit a proof of Diaconis and Saloff-Coste (1994…
This document contains supplementary material for the main articles in our Random Cayley Graphs project. We prove refined results about simple random walks on the integers and on the cycle. We are primarily interested in the entropy of these random walks at certain times and how this entropy changes when the time changes slightly. Additionally, we prove some large deviation and exit time estimates. We prove some results on the size of discrete lattice balls and how this size changes when the radius changes slightly. We do this in a general $L_q$ norm, with $q \in [1,\infty]$. We also prove som…
Consider continuous-time random walks on Cayley graphs where the rate assigned to each edge depends only on the corresponding generator. We show that the limiting speed is monotone increasing in the rates for infinite Cayley graphs that arise from Coxeter systems, but not for all Cayley graphs. On finite Cayley graphs, we show that the distance -- in various senses -- to stationarity is monotone decreasing in the rates for Coxeter systems and for abelian groups, but not for all Cayley graphs. We also find several examples of surprising behaviour in the dependence of the distance to stationarit…
In these expository notes, we describe some features of the multiplicative coalescent and its connection with random graphs and minimum spanning trees. We use Pitman's proof of Cayley's formula, which proceeds via a calculation of the partition function of the additive coalescent, as motivation and as a launchpad. We define a random variable which may reasonably be called the empirical partition function of the multiplicative coalescent, and show that its typical value is exponentially smaller than its expected value. Our arguments lead us to an analysis of the susceptibility of the Erdős-Rény…
Graph-theoretical estimates of the diameters of the Rubik's Cube groups
本文提出一种基于图阶数、度数、围长及过顶点圈数等局部参数的对称图直径严格下界方法，并应用于魔方群凯莱图，获得比随机正则图下界更紧、达到正确直径60%至77%的估计。
We show that the lamplighter group L has a system of generators for which the spectrum of the discrete Laplacian on the Cayley graph is a union of an interval and a countable set of isolated points accumulating to a point outside this interval. This is the first example of a group with infinitely many gaps in the spectrum of its Cayley graph. The result is obtained by a careful study of spectral properties of a one-parametric family of convolution operators on L. Our results show that the spectrum is a pure point spectrum for each value of the parameter, the eigenvalues are solutions of algebr…
On the diameter of permutation groups
给定一个有限群$G$和一个生成元集合$A$，Cayley图$Γ(G,A)$的直径$(Γ(G,A))$是使$G$的每个元素都能表示为$A \cup A^{-1}$中长度至多为$\ell$的字的最小$\ell$.我们关心的是有界diam（G）：= $\max_A$ diam $(Γ(G,A))$。长期以来，人们一直认为$n$次对称群的直径在$n$中是多项式有界的，但在$\sqrt{n \log n}$中已知的最好的上界是指数的.我们给出了一个拟多项式上界，即当G = Sym（n）或G = \Alt（n）时的\[\text{diam}(G) = \exp(O((\log n)^4 \log\log n)) = \exp((\log \log |G|)^{O(1)})\]，其中隐含常数是绝对的.这解决了一个关键的公开案件的Babai猜想直径的简单群体。通过标准结果，我们的界也意味着一个拟多项式上界的直径的所有传递置换群的程度PDMATHAMT。
Geometry of Random Cayley Graphs of Abelian Groups
研究了有限阿贝尔群随机Cayley图的几何性质，证明了图距离和直径渐近集中且仅依赖群规模与生成元数，并推广了Alon-Roichman谱隙结果。
On the clique number of random Cayley graphs and related topics
本文用纯组合方法证明随机凯莱图的团数高概率为 O(log N log log N)，改进了 Alon 的上界，并据此证明 Alon 关于凯莱图团数与独立数均为 O(log N) 的猜想对几乎所有阿贝尔群成立，还构造了自补凯莱图以强形式解决 Alon–Orlitsky 问题。
Given a finite group, we study the Gaussian series of the matrices in the image of its left regular representation. We propose such random matrices as a benchmark for improvements to the noncommutative Khintchine inequality, and we highlight an application to the matrix Spencer conjecture.
In the article random functions in modules over the octonion algebra and Cayley-Dickson algebras are investigated. For their study transition measures with values in the octonion algebra and Cayley-Dickson algebras are used. Stochastic integrals over these algebras are studied. They are applied to integration of stochastic PDEs. This approach permits subsequently to analyze and integrate PDEs of orders higher than two of different types including parabolic, elliptic and hyperbolic.
Elephant random walks on infinite Cayley trees
本文将Schütz和Trimper的大象随机游走推广到有限生成群，研究Cayley树的非交换群的渐近速度，发现速度与记忆参数无关，且存在相变点。
Our main result is an extension of Pansu's theorem to random metrics, where the edges of the Cayley are i.i.d. random variable with some finite exponential moment. Based on a previous work by the second author, the proof relies on Talagrand's concentration inequality, and on Pansu's theorem. Adapting a well-known argument for Z^d, we prove a sublinear estimate on the variance for virtually nilpotent groups which are not virtually isomorphic to Z. We further discuss the asymptotic cones of first-passage percolation on general infinite connected graphs: we prove that the asymptotic cones are a.e…
For a transitive infinite connected graph $G$, let $μ(G)$ be its connective constant. Denote by $\mathbf{\cal G}$ the set of Cayley graphs for finitely generated infinite groups with an infinite-order generator which is independent of other generators. Assume $G\in\mathbf{\cal G}$ is a Cayley graph of a finitely presented group, and Cayley graph sequence $\{G_n\}_{n=1}^{\infty}\subset \mathbf{\cal G}$ converges locally to $G.$ Then $μ(G_n)$ converges to $μ(G)$ as $n\rightarrow\infty.$ This confirms partially a conjecture raised by Benjamini [2013. {\it Coarse geometry and randomness.} Lect. No…
Recently, there has been significant interest in understanding the properties of Markov random fields (M.r.f.) defined on the independent sets of sparse graphs. When these M.r.f. are restricted to pairwise interactions (i.e. hardcore model), much progress has been made. However, considerably less is known in the presence of higher-order interactions, which arise e.g. in the analysis of independent sets with special properties and the study of resource-constrained communication networks. In this paper, we further our understanding of such models by analyzing M.r.f. with second-order interaction…
Central limit theorems for linear statistics of lattice random fields (including spin models) are usually proven under suitable mixing conditions or quasi-associativity. Many interesting examples of spin models do not satisfy mixing conditions, and on the other hand, it does not seem easy to show central limit theorem for local statistics via quasi-associativity. In this work, we prove general central limit theorems for local statistics and exponentially quasi-local statistics of spin models on discrete Cayley graphs with polynomial growth. Further, we supplement these results by proving simil…
Cutoff for random Cayley graphs of nilpotent groups
研究秩和幂零类有界的有限幂零群随机Cayley图上简单随机游走的截止现象，证明随机生成元情形下高概率出现截止，并给出一般生成元集合下谱隙和混合时间与阿贝尔化投影渐近等价的判据。
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…
Sensitivity of mixing times of Cayley graphs
本文证明全变差混合时间在拟等距下不是不变的，通过构造度数无界的 Cayley 图对及有界度非传递图，展示混合时间之比可趋于无穷大。
Percolation Perturbations in Potential Theory and Random Walks
我们证明了在一个非顺从群的Cayley图上，Bernoulli渗流的无限簇对于简单随机游动几乎必然是瞬态的，在这些簇上的简单随机游动具有正速度，并且这些簇允许有界的调和函数.一个主要的新发现，这些结果的基础是，这样的集群承认不变的随机子图正等周常数。我们还表明，在任何顺从凯莱图的渗流集群几乎肯定不承认非常数调和狄利克雷函数。相反，在一个含有非常数调和Dirichlet函数的Cayley图上，当$p$充分接近1时，几乎必然$p$ -Bernoulli渗流的无限簇也含有非常数调和Dirichlet函数.提出了许多问题和建议。
On the trace of branching random walks
研究Cayley图上的分支随机游动。第一个结果是Cayley图上的瞬时分支随机游动的迹是a. s。对于简单随机游动是瞬态的。此外，它还拥有A. S。临界逾渗概率小于1，体积指数增长。证明依赖于这样一个事实，即迹诱导的分支随机游动的家庭树不变的渗流。此外，我们证明了迹是a.s.对于任何（非平凡）分支随机游动都是强返回的。这是从迹的观察得出的，在适当偏置根之后，迹定义了么模测度。所有的结果表明，在更一般的情况下，分支随机游动的单模随机图。
We determine the asymptotic growth rate of the diameter of the random hyperbolic surfaces constructed by Brooks and Makover. This model consists of a uniform gluing of $2n$ hyperbolic ideal triangles along their sides followed by a compactification to get a random hyperbolic surface of genus roughly $n/2$. We show that the diameter of those random surfaces is asymptotic to $2 \log n$ in probability as $n \to \infty$.
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