Bernardo N. B. de Lima, Rémy Sanchis, Roger W. C. Silva
Consider an independent site percolation model on $\Z^d,\ d\geq 2$, with parameter $p \in (0,1)$, where there are only nearest neighbor bonds and long range bonds of length $k$ parallel to some coordinate axis. We show that the percolation probability is a non-decreasing function of $k$. We also generalize this result for models whose long range bonds have several lengths.
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We consider standard first-passage percolation on $\Z^d$. Let $e_1$ be the first coordinate vector. Let $a(n)$ be the expected passage time from the origin to $ne_1$. In this short paper, we note that $a(n)$ is increasing under some strong condition on the support of the law of the passage times on the edges.
This paper studies the first passage percolation (FPP) model: each edge in the cubic lattice is assigned a random passage time, and consideration is given to the behavior of the percolation region $B(t)$, which consists of those vertices that can be reached from the origin within a time $t > 0$. Cox and Durrett showed the shape theorem for the percolation region, saying that the normalized region $B(t)/t$ converges to some limit shape $\mathcal{B}$. This paper introduces a general FPP model defined on crystal lattices, and shows the monotonicity of the limit shapes under covering maps, thereby…
We consider Bernoulli bond percolation on oriented regular trees, where besides the usual short bonds, all bonds of a certain length are added. Independently, short bonds are open with probability $p$ and long bonds are open with probability $q$. We study properties of the critical curve which delimits the set of pairs $(p,q)$ for which there are almost surely no infinite paths. We also show that this curve decreases with respect to the length of the long bonds.
Consider an anisotropic independent bond percolation model on the $d$-dimensional hypercubic lattice, $d\geq 2$, with parameter $p$. We show that the two point connectivity function $P_{p}(\{(0,\dots,0)\leftrightarrow (n,0,\dots,0)\})$ is a monotone function in $n$ when the parameter $p$ is close enough to 0. Analogously, we show that truncated connectivity function $P_{p}(\{(0,\dots,0)\leftrightarrow (n,0,\dots,0), (0,\dots,0)\nleftrightarrow\infty\})$ is also a monotone function in $n$ when $p$ is close to 1.
Monotonicity of the critical point in two-dimensional oriented percolation with enhancement
本文研究了二维定向渗流中临界点的单调性，通过引入对角线边的概率参数ε，证明了临界参数随ε的增加而严格递减。
In 1993 van den Berg and Kesten proved a strict monotonicity theorem for first passage percolation on $\mathbb{Z}^d$, $d \ge 2$: given two probability measures $ν$ and $\tildeν$ with finite mean, if $\tildeν$ is strictly more variable than $ν$ and $ν$ is subcritical in an appropriate sense, the time constant associated to $\tildeν$ is strictly smaller than the time constant associated to $ν$. In this paper, an analogous result is proven for (not necessarily almost-transitive) graphs of strict polynomial growth and for bounded degree graphs quasi-isometric to trees which satisfy a certain geome…
Strict monotonicity of critical points in independent long-range percolation models
该研究针对局部有限顶点传递图上的独立长程渗流模型，通过耦合思想证明临界点关于连接函数局部扰动的严格单调性，在更少假设下改进了经典方法的结果，且适用于无向和有向边渗流模型。
We consider anisotropic independent bond percolation models on the slab $\Z^2\times\{0,\dots,k\}$, where we suppose that the axial (vertical) bonds are open with probability $p$, while the radial (horizontal) bonds are open with probability $q$. We study the critical curves for these models and establish their continuity and strict monotonicity.
A useful result about leftmost and rightmost paths in two dimensional bond percolation is proved. This result was introduced without proof in \cite{G} in the context of the contact process in continuous time. As discussed here, it also holds for several related models, including the discrete time contact process and two dimensional site percolation. Among the consequences are a natural monotonicity in the probability of percolation between different sites and a somewhat counter-intuitive correlation inequality.
We answer a question of Benjamini and Schramm by proving that under reasonable conditions, quotienting a graph strictly increases the value of its percolation critical parameter $p_c$. More precisely, let $\mathcal{G}=(V,E)$ be a quasi-transitive graph with $p_c(\mathcal{G})<1$, and let $G$ be a nontrivial group that acts freely on $V$ by graph automorphisms. Assume that $\mathcal{H}:=\mathcal{G}/G$ is quasi-transitive. Then one has $p_c(\mathcal{G})<p_c(\mathcal{H})$. We provide results beyond this setting: we treat the case of general covering maps and provide a similar result for the unique…
Monotonicity properties for Bernoulli percolation on layered graphs -- a Markov chain approach
针对分层图上伯努利渗流的连通概率单调性猜想，提出逐层构建渗流构型的马尔可夫链方法，证明有限基图下该单调性对足够大的层差成立，为相关渗流问题提供新研究思路。
The critical curve of long-range percolation on oriented trees
该研究在有向树上分析长程渗透模型，通过特征多项式研究得到临界曲线的性质及新界，揭示其连续性和凸性特征。
The probability of connection between two vertices cannot be monotone with the distance for Bernoulli Percolation on transitive graphs
本文研究了伯努利渗流中两个顶点连接概率与距离的关系，发现其不必然随距离单调递减，并定义了临界点现象。
Percolation in the marked stationary Random Connection Model for higher-dimensional simplicial complexes
本研究提出将经典随机连接模型推广到随机单纯复形的新型渗流模型，证明了上连通性渗流中巨分量出现的锐相变，还得到多个经典复形模型的锐相变特例，明确了这类渗流模型的必要性质。
Using recent couplings we provide counterexamples to monotonicity properties of percolation models related to graphical representations of the Ising model. We further prove a new coupling of the double random current model to the loop-$\mathrm{O}(1)$-model.
We study a general class of percolation models in Euclidean space including long-range percolation, scale-free percolation, the weight-dependent random connection model and several other previously investigated models. Our focus is on the weak decay regime, in which inter-cluster long-range connection probabilities fall off polynomially with small exponent, and for which we establish several structural properties. Chief among them are the continuity of the bond percolation function and the transience of infinite clusters.
Consider critical site percolation on $\mathbb{Z}^d$ with $d \geq 2$. We prove a lower bound of order $n^{- d^2}$ for point-to-point connection probabilities, where $n$ is the distance between the points. Most of the work in our proof concerns a `construction' which finally reduces the problem to a topological one. This is then solved by applying a topological fact, which follows from Brouwer's fixed point theorem. Our bound improves the lower bound with exponent $2 d (d-1)$, used by Cerf in 2015 to obtain an upper bound for the so-called two-arm probabilities. Apart from being of interest in …
Monotonicity of Markov chain transition probabilities via quasi-stationarity -- an application to Bernoulli percolation on $C_k \times Z$
该研究通过马尔可夫链准平稳分布的收敛速率界得到转移概率单调性阈值的显式界，并将其应用于柱图伯努利渗流，证明了连通概率的一致单调性结果。
We consider the supercritical finite-range random connection model where the points $x,y$ of a homogeneous planar Poisson process are connected with probability $f(|y-x|)$ for a given $f$. Performing percolation on the resulting graph, we show that the critical probabilities for site and bond percolation satisfy the strict inequality $p_c^{\rm site} > p_c^{\rm bond}$. We also show that reducing the connection function $f$ strictly increases the critical Poisson intensity. Finally, we deduce that performing a spreading transformation on $f$ (thereby allowing connections over greater distances b…
Percolation phase transition in weight-dependent random connection models
研究权重依赖随机连接模型中的渗流相变，发现其由度分布幂律指数和几何参数共同决定，并应用于刻画基于年龄的空间优先连接网络的鲁棒性。
Percolation in a hierarchical random graph
我们研究了嵌入在阶为$N$的层次群中的无限随机图${\cal G}_N$中的渐近渗流为$N\to \infty$，连通概率依赖于顶点之间的超度量距离. ${\cal G}_N$被构造为有限随机子图的级联（近似）Erdös-Renyi型.我们给出了一个标准的渗透，并表明渗透发生沿巨组件的巨组件在前一级的级联的子图的所有连续的层次距离。证明涉及一个层次的随机图的顶点具有内部结构和随机连接概率。
Crossing bridges between percolation models and Bienaymé-Galton-Watson trees
本文综述了渗流理论与种群遗传模型之间的联系，重点介绍了Galton-Watson树上的构造及Divide-and-Color渗流模型与多类型Galton-Watson树的新颖关联。
Let $G_{n,p}^1$ be a superposition of the random graph $G_{n,p}$ and a one-dimensional lattice: the $n$ vertices are set to be on a ring with fixed edges between the consecutive vertices, and with random independent edges given with probability $p$ between any pair of vertices. Bootstrap percolation on a random graph is a process of spread of "activation" on a given realisation of the graph with a given number of initially active nodes. At each step those vertices which have not been active but have at least $r \geq 2$ active neighbours become active as well. We study the size of the final act…
Sharpness of the percolation phase transition for weighted random connection models
本文研究加权随机连接模型中渗透相变的锐度，通过证明亚临界和超临界区域的簇大小分布特性，揭示了临界行为的数学结构。
A finite-graph conjecture related to $θ(p_c)=0$ for Bernoulli bond percolation on $\mathbb Z^d$
该研究提出有限图伯努利键渗流的一个猜想，可推出d≥2时Z^d上θ(p_c)=0，证明有限平面图情形成立，反例显示连通性结构不可或缺。
Two vertices are said to be finitely connected if they belong to the same cluster and this cluster is finite. We derive sharp asymptotics for finite connection probabilities for supercritical Bernoulli bond percolation on Z^2.
Existence of subcritical percolation phases for generalised weight-dependent random connection models
本文研究带标记泊松点过程上的广义权重依赖随机连接模型，通过控制长边给出临界环带穿越强度为正及次临界相存在的充分条件，并建立典型簇直径与规模的尾界。
In this paper we establish a connection between epidemic models on random networks with general infection times considered in Barbour and Reinert 2013 and first passage percolation. Using techniques developed in Bhamidi, van der Hofstad, Hooghiemstra 2012, when each vertex has infinite contagious periods, we extend results on the epidemic curve in Barbour Reinert 2013 from bounded degree graphs to general sparse random graphs with degrees having finite third moments as the number of vertices tends to infinity. We also study the epidemic trail between the source and typical vertices in the grap…
We exploit a connection between distances in the infinite percolation cluster, when the parameter is close to one, and the discrete-time TASEP on $\mathbb{Z}$. This shows that when the parameter goes to one, large balls in the cluster are asymptotically shaped near the axes like arcs of parabola.
Largest component and sharpness in continuum percolation
本文研究了在非临界情况下泊松随机连接模型中大连接组件的行为，证明了在亚临界情况下最大组件大小在窗口体积上以对数增长，在超临界情况下线性增长，并证明了在亚临界情况下原点集群的尾部呈指数衰减。
We use the connection between bond percolation and SIR epidemics to establish lower bounds for the critical percolation probability in $2$ and $3$ dimensions as the range becomes large. The bound agrees with the conjectured asymptotics for the long range critical probability, refines results of M. Penrose, and complements results of van der Hofstad and Sakai in dimensions greater than $6$.
We study the effects of mobility on two crucial characteristics in multi-scale dynamic networks: percolation and connection times. Our analysis provides insights into the question, to what extent long-time averages are well-approximated by the expected values of the corresponding quantities, i.e., the percolation and connection probabilities. In particular, we show that in multi-scale models, strong random effects may persist in the limit. Depending on the precise model choice, these may take the form of a spatial birth-death process or a Brownian motion. Despite the variety of structures that…
We study phase transition and percolation at criticality for three random graph models on the plane, viz., the homogeneous and inhomogeneous enhanced random connection models (RCM) and the Poisson stick model. These models are built on a homogeneous Poisson point process $\mathcal{P}_λ$ in $\mathbb{R}^2$ of intensity $λ$. In the homogenous RCM, the vertices at $x,y$ are connected with probability $g(|x-y|)$, independent of everything else, where $g:[0,\infty) \to [0,1]$ and $| \cdot |$ is the Euclidean norm. In the inhomogenous version of the model, points of $\mathcal{P}_λ$ are endowed with w…
We consider generic finite range percolation models on $\mathbb{Z}^d$ under a high temperature assumption (exponential decay of connection probabilities and exponential ratio weak mixing). We prove that the rate of decay of point-to-point connections exists in every directions and show that it naturally extends to a norm on $\mathbb{R}^d$. This result is the base input to obtain fine understanding of the high temperature phase and is usually proven using correlation inequalities (such as FKG). The present work makes no use of such model specific properties.
We prove a local limit theorem for the probability of a site to be connected by disjoint paths to three points in subcritical Bernoulli percolation on $\mathbb{Z}^{d},\,d\geq2$ in the limit where their distances tend to infinity.
Consider random sequential adsorption on a chequerboard lattice with arrivals at rate $1$ on light squares and at rate $λ$ on dark squares. Ultimately, each square is either occupied, or blocked by an occupied neighbour. Colour the occupied dark squares and blocked light sites {\em black}, and the remaining squares {\em white}. Independently at each meeting-point of four squares, allow diagonal connections between black squares with probability $p$; otherwise allow diagonal connections between white squares. We show that there is a critical surface of pairs $(λ, p)$, containing the pair $(1,0.…
Mean-field behaviour of the random connection model on hyperbolic space
本文研究了 $d=2,3$ 维双曲空间上的随机连接模型，证明了随强度 $λ$ 变化存在渗滤相变，并通过临界渗滤簇的等周性质推导出与平均场值一致的临界指数。
We prove that for Voronoi percolation on $\mathbb{R}^d$, there exists $p_c\in[0,1]$ such that - for $p<p_c$, there exists $c_p>0$ such that $\mathbb{P}_p[0\text{ connected to distance }n]\leq \exp(-c_p n)$, - there exists $c>0$ such that for $p>p_c$, $\mathbb{P}_p[0\text{ connected to }\infty]\geq c(p-p_c)$. For dimension 2, this result offers a new way of showing that $p_c(2)=1/2$. This paper belongs to a series of papers using the theory of algorithms to prove sharpness of the phase transition.
Phase transitions for percolation of words in one dimension
本文研究带长程连接的正整数集上的单词渗流模型，针对随机范围序列和非降确定性范围序列两种情形，均建立了从原点同时发生所有单词渗流的相变。
Consider an independent site percolation model on $\Z^d$, with parameter $p \in (0,1)$, where all long range connections in the axes directions are allowed. In this work we show that given any parameter $p$, there exists and integer $K(p)$ such that all binary sequences (words) $ξ\in \{0,1\}^{\N}$ can be seen simultaneously, almost surely, even if all connections whose length is bigger than $K(p)$ are suppressed. We also show some results concerning the question how $K(p)$ should scale with $p$ when $p$ goes to zero. Related results are also obtained for the question of whether or not almost a…
We study Bernoulli bond percolation on a random recursive tree of size $n$ with percolation parameter $p(n)$ converging to $1$ as $n$ tends to infinity. The sizes of the percolation clusters are naturally stored in a tree. We prove convergence in distribution of this tree to the genealogical tree of a continuous-state branching process in discrete time. As a corollary we obtain the asymptotic sizes of the largest and next largest percolation clusters, extending thereby a recent work of Bertoin (2014) which deals with cluster sizes in the supercritical regime. In a second part, we show that the…
Decay of connection probability in high-dimensional continuum percolation
研究高维随机连接模型中临界两点连接概率的衰减行为，利用花边展开和反卷积策略证明其衰减速度为|x|^{-(d-2)}，并简化了Hara在2008年的证明。
Finiteness of the percolation threshold for inhomogeneous long-range models in one dimension
该研究针对一维非均匀长程空间随机图，引入系数$\delta_\text{eff}$量化重尾度对长程连接的影响，证明了渗流阈值存在的临界条件，补充了已有相关研究结果，并为有限图巨分支存在性提供判定准则。
Deriving Moments in the Age of Gossip Process from Percolation
本文基于信息年龄与首达渗流的联系，利用概率论基本事实，更简洁地推导出Gossip网络中年龄过程任意k阶矩的递归恒等式，并推广了统计物理技术及指出与Eden模型的联系。
Percolation on random 2-lifts
本文研究了图$G$的随机2-提升模型上的渗流问题，证明了临界参数的连续性、严格单调性以及次临界状态下簇大小的指数衰减，为检验渗流理论证明方法的稳健性提供了依据。
The Busemann Process and Steep Highways in Directed First Passage Percolation
该研究拓展技术证明平面有向首次通过渗流中Busemann过程的存在性，得到可积模型下的显式分布，可用于量化半无限测地线与聚类现象，并得到离散时间并行更新TASEP的多类不变分布。
Recurrence of strong-decay inhomogeneous long-range percolation clusters
该研究证明了一维和二维非均匀长程渗流的重发性准则，推导了二维平面随机几何图满足重发性的条件，并得出带插值核的二维权重依赖随机连接模型在特定强衰减区域内连通分量为重发的结论。
We consider invasion percolation on the randomly-weighted complete graph $K_n$, started from some number $k(n)$ of distinct source vertices. The outcome of the process is a forest consisting of $k(n)$ trees, each containing exactly one source. Let $M_n$ be the size of the largest tree in this forest. Logan, Molloy and Pralat (arXiv:1806.10975) proved that if $k(n)/n^{1/3} \to 0$ then $M_n/n \to 1$ in probability. In this paper we prove a complementary result: if $k(n)/n^{1/3} \to \infty$ then $M_n/n \to 0$ in probability. This establishes the existence of a phase transition in the structure of…
Exponential decay for the random connection model using asymptotic transitivity
本研究推广Vanneuville的渗流方法，利用极高强度RCM的几乎顶点传递性，在极小假设下证明支撑集无界的次临界泊松RCM原点簇大小概率呈指数衰减，填补了该场景下的研究空白。
We study the behavior of the random walk in a continuum independent long-range percolation model, in which two given vertices $x$ and $y$ are connected with probability that asymptotically behaves like $|x-y|^{-α}$ with $α>d$, where $d$ denotes the dimension of the underlying Euclidean space. More precisely, focus is on the random connection model in which the vertex set is given by the realization of a homogeneous Poisson point process. We show that this random graph exhibits the same properties as classical discrete long-range percolation models studied in [3] with regard to recurrence and t…
Boolean percolation on digraphs and random exchange processes
本文在一般图论框架下研究 Lamperti 引入的长程渗流模型，探讨其与随机交换过程的联系，并阐明了格 $\mathbb{N}_0^n$ 和 $\mathbb{Z}^n$ 被本质覆盖的条件，同时证明无法覆盖有向 $n$ 叉树。
Only long edges can erase the subcritical annulus-crossing phase of weight-dependent random connection models
本文针对权重依赖随机连接模型的环带穿越次临界相问题，提出更直接且适用于经典模型的长边判据，弥补既有判据在部分情形下不确定的不足。
Chaos, concentration and multiple valleys in first-passage percolation
本文在首达渗流中建立了涨落与混沌行为之间的严格联系，证明轻微重采样会使时间最小化路径发生显著变化，并揭示混沌行为导致大量几乎不相交的近似最优路径（多重谷）存在。
Iterated localization is considered where each node of a network needs to get localized (find its location on 2-D plane), when initially only a subset of nodes have their location information. The iterated localization process proceeds as follows. Starting with a subset of nodes that have their location information, possibly using global positioning system (GPS) devices, any other node gets localized if it has three or more localized nodes in its radio range. The newly localized nodes are included in the subset of nodes that have their location information for the next iteration. This process …
In this paper we study the connection between the phenomenon of homological percolation (the formation of "giant" cycles in persistent homology), and the zeros of the expected Euler characteristic curve. We perform an experimental study that covers four different models: site-percolation on the cubical and permutahedral lattices, the Poisson-Boolean model, and Gaussian random fields. All the models are generated on the flat torus $T^d$, for $d=2,3,4$. The simulation results strongly indicate that the zeros of the expected Euler characteristic curve approximate the critical values for homologic…
Percolation in the Stochastic Block Model
本文研究社区结构随网络规模演化的随机块模型序列中的渗流，发现临界平均度通常不等于1，并提出了社区尺度分支过程方法以准确预测相变，扩展了渗流研究至更现实场景。
Dropout Neural Network Training Viewed from a Percolation Perspective
本文研究使用dropout训练深度神经网络时的逾渗现象，建立新逾渗模型刻画网络拓扑与路径问题的关系，揭示dropout中的逾渗效应及其可能导致训练崩溃的机制。
We obtain tight thresholds for bond percolation on one-dimensional small-world graphs, and apply such results to obtain tight thresholds for the \emph{Independent Cascade} process and the \emph{Reed-Frost} process in such graphs. These are the first fully rigorous results establishing a phase transition for bond percolation and SIR epidemic processes in small-world graphs. Although one-dimensional small-world graphs are an idealized and unrealistic network model, a number of realistic qualitative epidemiological phenomena emerge from our analysis, including the epidemic spread through a sequen…
We prove Ornstein-Zernike behaviour in every direction for finite connection functions of bond percolation on $\mathbb{Z}^{d}$ for $d\geq3$ when $p,$ the probability of occupation of a bond, is sufficiently close to $1.$ Moreover, we prove that equi-decay surfaces are locally analytic, strictly convex, with positive Gaussian curvature.
Inspired by biological evolution, we consider the following so-called accessibility percolation problem: The vertices of the unoriented $n$-dimensional binary hypercube are assigned independent $U(0, 1)$ weights, referred to as fitnesses. A path is considered accessible if fitnesses are strictly increasing along it. We prove that the probability that the global fitness maximum is accessible from the all zeroes vertex converges to $1-\frac{1}{2}\ln\left(2+\sqrt{5}\right)$ as $n\rightarrow\infty$. Moreover, we prove that if one conditions on the location of the fitness maximum being $v$, then pr…
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