Percolation, boundary, noise: an experiment
Boris Tsirelson
临界渗流的标度极限是黑噪声吗？答案取决于对沿直线集中的扰动的稳定性。这段文字，不包含证据，报告实验结果，建议肯定的答案。
The scaling limit of the critical percolation, is it a black noise? The answer depends on stability to perturbations concentrated along a line. This text, containing no proofs, reports experimental results that suggest the affirmative answer.
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We state and prove a version for invasion percolation of Schramm and Smirnov's (2001) discrete gluing theorem for critical planar percolation. The result is a first step toward establishing factorization and characterizing noise sensitivity for invasion percolation.
Convergence of Half-Space Last Passage Percolation Away from the Boundary to the Directed Landscape
本文研究了半空间指数最后通道模型在边界外处对定向景观的收敛性，通过耦合半空间与全空间模型并结合中等偏差估计，证明了其收敛性。
We study frozen percolation on the (planar) triangular lattice, where connected components stop growing ("freeze") as soon as their "size" becomes at least $N$, for some parameter $N \geq 1$. The size of a connected component can be measured in several natural ways, and we consider the two particular cases of diameter and volume (i.e. number of sites). Diameter-frozen and volume-frozen percolation have been studied in previous works, and they display radically different behaviors. These works adopt the rule that the boundary of a frozen cluster stays vacant forever, and we investigate the infl…
We prove the existence of phase transitions in the global connectivity of the excursion sets of planar symmetric shot noise fields. Our main result establishes a phase transition with respect to the level for shot noise fields with symmetric log-concave mark distributions, including Gaussian, uniform, and Laplace marks, and kernels that are positive, symmetric, and have sufficient tail decay. Without the log-concavity assumption we prove a phase transition with respect to the intensity of positive marks.
Noise sensitivity in last-passage percolation
本文研究了Kardar-Parisi-Zhang普遍性类中空间生长过程的噪声敏感性，证明了几何最后一层渗透中旅行时间的噪声敏感性。
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。
Loop soup percolation and boundary dimension on metric trees
该研究确定了具有适当电缆度量的暂态局部有限加权树上布朗环汤的渗流阈值，给出了伯努利渗流 Lyons 公式的类似物，并将分类扩展至超临界 Galton-Watson 树及相关实树。
Noise sensitivity in bootstrap percolation
本文讨论了Itai Benjamini的问题，证明了在d-维盒[n]^d上的2-近邻自举渗流中，对于\geq 2，在其临界初始密度p_c（n），完全占据事件是噪声敏感的，而在d-正则随机图G_{n,d}上的k-近邻自举渗流中，对于2 \leq k \leq d-2，完全占据事件是不敏感的.许多悬而未决的问题仍然存在。
A large and sparse random graph with independent exponentially distributed link weights can be used to model the propagation of messages or diseases in a network with an unknown connectivity structure. In this article we study an extended setting where also the nodes of the graph are equipped with nonnegative random weights which are used to model the effect of boundary delays across paths in the network. Our main results provide approximative formulas for typical first passage times, typical flooding times, and maximum flooding times in the extended setting, over a time scale logarithmic with…
We consider first-passage percolation on the two-dimensional triangular lattice $\mathcal{T}$. Each site $v\in\mathcal{T}$ is assigned independently a passage time of either $0$ or $1$ with probability $1/2$. Denote by $B^+(0,n)$ the upper half-disk with radius $n$ centered at $0$, and by $c_n^+$ the first-passage time in $B^+(0,n)$ from $0$ to the half-circular boundary of $B^+(0,n)$. We prove \[\lim_{n\rightarrow\infty}\frac{c_n^+}{\log n}=\frac{\sqrt{3}}{2π}~ a.s.,~\lim_{n\rightarrow\infty}\frac{E c_n^+}{\log n}=\frac{\sqrt{3}}{2π},~\lim_{n\rightarrow\infty}\frac{\mathrm{Var}(c_n^+)}{\log n…
Degenerate Diffusions on Continuum Percolation and Hamilton-Jacobi-Bellman Equations
研究在连续渗流簇上的退化受控扩散和Hamilton-Jacobi-Bellman方程，证明扩散永不触及边界并存在唯一全局强解，建立随机控制表示公式，验证布尔模型等具体模型满足假设。
First-passage percolation is a random growth model defined using i.i.d. edge-weights $(t_e)$ on the nearest-neighbor edges of $\mathbb{Z}^d$. An initial infection occupies the origin and spreads along the edges, taking time $t_e$ to cross the edge $e$. In this paper, we study the size of the boundary of the infected ("wet") region at time $t$, $B(t)$. It is known that $B(t)$ grows linearly, so its boundary $\partial B(t)$ has size between $ct^{d-1}$ and $Ct^d$. Under a weak moment condition on the weights, we show that for most times, $\partial B(t)$ has size of order $t^{d-1}$ (smooth). On th…
In this paper we study noise sensitivity and threshold phenomena for Poisson Voronoi percolation on $\mathbb{R}^2$. In the setting of Boolean functions, both threshold phenomena and noise sensitivity can be understood via the study of randomized algorithms. Together with a simple discretization argument, such techniques apply also to the continuum setting. Via the study of a suitable algorithm we show that box-crossing events in Voronoi percolation are noise sensitive and present a threshold phenomenon with polynomial window. We also study the effect of other kinds of perturbations, and emphas…
Spectra of Poisson functionals and applications in continuum percolation
本文为泊松泛函引入谱点过程概念，并利用其证明临界连续渗流模型（泊松布尔与泊松沃罗诺伊）中交叉事件的尖锐噪声不稳定性和敏感性，核心工具是 Wiener-Itô 展开与拟乘性估计。
Lectures on noise sensitivity and percolation
本文提供了在2010年巴西布齐奥斯粘土暑期学校的课程“噪声敏感性和渗透”的讲义。
Last passage percolation and traveling fronts
我们考虑一个由Brunet和Derrida引入的具有随机动力学的N粒子系统。粒子可以被解释为在{1，.，N}的平均场类型。粒子保持分组并像行波一样移动，受到离散化并由随机噪声驱动。随着N的增加，我们获得了对不同的驾驶噪声规律的前部速度及其轮廓的估计。的Gumbel分布起着核心作用的粒子跳跃，我们表明，在这种情况下，标度极限是一个Lévy过程。有界跳跃的情况产生完全不同的行为。
Let $d\geq 2$. We consider an i.i.d. supercritical bond percolation on $\mathbb{Z}^d$, every edge is open with a probability $p>p_c(d)$, where $p_c(d)$ denotes the critical point. We condition on the event that $0$ belongs to the infinite cluster $\mathcal{C}_\infty$ and we consider connected subgraphs of $\mathcal{C}_\infty$ having at most $n^d$ vertices and containing $0$. Among these subgraphs, we are interested in the ones that minimize the open edge boundary size to volume ratio. These minimizers properly rescaled converge towards a translate of a deterministic shape and their open edge b…
Consider critical Bernoulli percolation in the plane. We give a new proof of the sharp noise sensitivity theorem shown by Garban, Pete and Schramm. Contrary to the previous approaches, we do not use any spectral tool. We rather study differential inequalities satisfied by a dynamical four-arm event, in the spirit of Kesten's proof of scaling relations. We also obtain new results in dynamical percolation. In particular, we prove that the Hausdorff dimension of the set of times with both primal and dual percolation equals $2/3$ a.s.
We study the isoperimetric subgraphs of the giant component $\textbf{C}_n$ of supercritical bond percolation on the square lattice. These are subgraphs of $\textbf{C}_n$ having minimal edge boundary to volume ratio. In contrast to the work of Biskup, Louidor, Procaccia and Rosenthal, the edge boundary is taken only within $\textbf{C}_n$ instead of the full infinite cluster. The isoperimetric subgraphs are shown to converge almost surely, after rescaling, to the collection of optimizers of a continuum isoperimetric problem emerging naturally from the model. We also show that the Cheeger constan…
Let $Q$ be a free Boltzmann quadrangulation with simple boundary decorated by a critical ($p=3/4$) face percolation configuration. We prove that the chordal percolation exploration path on $Q$ between two marked boundary edges converges in the scaling limit to chordal SLE$_6$ on an independent $\sqrt{8/3}$-Liouville quantum gravity disk (equivalently, a Brownian disk). The topology of convergence is the Gromov-Hausdorff-Prokhorov-uniform topology, the natural analog of the Gromov-Hausdorff topology for curve-decorated metric measure spaces. We also obtain analogous scaling limit results for fa…
Noise Sensitivity in Continuum Percolation
我们证明了泊松布尔模型，也被称为吉尔伯特盘模型，是噪声敏感的临界。这是连续渗流模型的第一个这样的结果，也是临界概率p_c \ne 1/2的第一个结果。我们的证明使用了一个版本的Benjamini-Kalai-Schramm定理有偏见的积测度。这个结果的一个定量版本最近被Keller和Kindler证明。我们给出了一个简单的推导的非定量结果的无偏版本。我们还开发了一个相当普遍的方法，近似连续渗流模型的离散模型与p_c有界远离零，这种方法是基于非均匀超图的极值结果。
We show that planar Bargmann-Fock percolation is noise sensitive under the Ornstein-Ulhenbeck process. The proof is based on the randomized algorithm approach introduced by Schramm and Steif and gives quantitative polynomial bounds on the noise sensitivity of crossing events for Bargmann-Fock. A rather counter-intuitive consequence is as follows. Let $F$ be a Bargmann-Fock Gaussian field in $\mathbb{R}^3$ and consider two horizontal planes $P_1,P_2$ at small distance $\varepsilon$ from each other. Even though $F$ is a.s. analytic, the above noise sensitivity statement implies that the full res…
Lower bounds for boundary roughness for droplets in Bernoulli percolation
我们认为边界粗糙度的“液滴”时创建的超临界二维伯努利渗流的条件下有一个开放的双重电路周围的起源和封闭的区域至少$l^2$，大$l$。最大的局部粗糙度是最大的向内偏差的液滴边界从它自己的凸包的边界，我们表明，对于大$l$这个最大值至少是为了$l^{1/3}(\log l)^{-2/3}$。这补充了已知的平均局部粗糙度的阶数$l^{1/3}(\log l)^{2/3}$的上界。这里$l$的指数为1/3，这与物理学文献对二维界面的预测一致。
We consider the Bernoulli percolation model in a finite box and we introduce an automatic control of the percolation probability, which is a function of the percolation configuration. For a suitable choice of this automatic control, the model is self-critical, i.e., the percolation probability converges to the critical point $p_c$ when the size of the box tends to infinity. We study here three simple examples of such models, involving the size of the largest cluster, the number of vertices connected to the boundary of the box, or the distribution of the cluster sizes.
An Exact Strip Observable and the Half-Plane One-Arm Probability for Critical Bond Percolation on the Square Lattice
本文证明了方格上临界伯努利键逾渗的半平面单臂指数，通过修正Ikhlef-Ponsaing插值并建立精确条带公式，结合RSW定理得到单臂概率的幂律衰减。
Directed Percolation in Wireless Networks with Interference and Noise
以前的无线网络连通性研究主要集中在无向几何图上。然而，更复杂的模型，如信号干扰噪声比（SINR）模型，通常会导致有向图。在本文中，我们研究了渗流过程中的无线网络建模的有向SINR图。我们首先研究无干扰网络，在那里我们定义了四种类型的相变，并表明它们同时发生。通过将有向SINR图与另外两个无向SINR图耦合，我们进一步得到了临界密度的解析上界和下界。然后，我们表明，干扰，渗流在有向SINR图不仅取决于密度，但也对逆系统的处理增益。我们还提供了逆系统处理增益的临界值的界限。
Invasion percolation is a stochastic growth model that follows a greedy algorithm. After assigning i.i.d. uniform random variables (weights) to all edges of $\mathbb{Z}^d$, the growth starts at the origin. At each step, we adjoin to the current cluster the edge of minimal weight from its boundary. In '85, Chayes-Chayes-Newman studied the `acceptance profile' of the invasion: for a given $p \in [0,1]$, it is the ratio of the expected number of invaded edges until time $n$ with weight in $[p,p+\text{d}p]$ to the expected number of observed edges (those in the cluster or its boundary) with weight…
We initiate the study on chemical distances of percolation clusters for level sets of two-dimensional discrete Gaussian free fields as well as loop clusters generated by two-dimensional random walk loop soups. One of our results states that the chemical distance between two macroscopic annuli away from the boundary for the random walk loop soup at the critical intensity is of dimension 1 with positive probability. Our proof method is based on an interesting combination of a theorem of Makarov, isomorphism theory and an entropic repulsion estimate for Gaussian free fields in the presence of a h…
We introduce the Incipient Infinite Cluster (IIC) in the critical Bernoulli site percolation model on the Uniform Infinite Half-Planar Triangulation (UIHPT), which is the local limit of large random triangulations with a boundary. The IIC is defined from the UIHPT by conditioning the open percolation cluster of the origin to be infinite. We prove that the IIC can be obtained by adding within the UIHPT an infinite triangulation with a boundary whose distribution is explicit.
We discuss duality properties of critical Boltzmann planar maps such that the degree of a typical face is in the domain of attraction of a stable distribution with parameter $α\in(1,2]$. We consider the critical Bernoulli bond percolation model on a Boltzmann map in the dilute and generic regimes $α\in (3/2,2]$, and show that the open percolation cluster of the origin is itself a Boltzmann map in the dense regime $α\in (1,3/2)$, with parameter \[α':= \frac{2α+3}{4α-2}.\] This is the counterpart in random planar maps of the duality property $κ\leftrightarrow 16/κ$ of Schramm-Loewner Evolutions …
Quantitative noise sensitivity and exceptional times for percolation
本文的目的之一是证明平面三角形格子上的动态临界点渗流存在例外的渗流时刻。在这样做时，新的定量逾渗噪声灵敏度的结果。后者基于一种新颖的方法，用于通过构造随机化算法来控制“k级”傅里叶系数，该随机化算法查看随机比特，输出特定函数的值，但查看具有低概率的任何固定输入比特。我们也得到了上，下界的豪斯多夫维数的时间集。然后，我们研究的问题，特殊的时间为某些“k臂”事件的楔和锥。作为这种分析的推论，我们证明，除其他事项外，有没有时间，有两个无限的“白色”集群，得到了无限白团簇和无限黑团簇同时存在的时间集合的豪斯多夫维数的上界，并证明了正方形网格上的动态临界键渗流不存在三个不相交的无限团簇同时存在的例外时间都在场
Percolation on random triangulations and stable looptrees
研究了Angel & Schramm的均匀无限平面三角剖分上的点渗流问题。我们计算了几个临界和近临界指数，并描述了在所有制度（亚临界，临界和超临界）的大渗流集群的边界的标度限制。我们特别证明了大临界渗流集团边界的标度极限是指数为3/2的随机稳定环树，这是在一个配套文件中介绍的。我们还提出了一个猜想，将（1，2）中任意指标的循环树与O（N）模型装饰的随机三角剖分中簇边界的标度极限联系起来。
Exclusion Sensitivity of Boolean Functions
近年来，布尔函数的噪声敏感性和噪声稳定性的研究受到了广泛的关注。本文的目的是将这些概念以一种自然的方式扩展到另一类扰动，即那些由运行对称排它过程短时间产生的扰动。在这项研究中，单调布尔函数的情况下将成为特别感兴趣的。我们证明了对于这类函数，普通的噪声敏感性和关于完全图排它过程的噪声敏感性是等价的。我们也证明了这种等价性的稳定性。在获得这些相当普遍的结果后，我们研究了“排斥敏感性”的临界逾渗更详细的中期动态。排斥动力学，由于其保守性，在某种意义上比经典的i.i.d动力学更物理化。有趣的是，我们将看到，为了获得对逾渗的排它敏感性的精确理解，我们将需要描述典型的逾渗谱集在潜在的排它过程下如何扩散。
Shape curvatures and transversal fluctuations in the first passage percolation model
本文考虑正方形格子上的第一次通过渗流模型。在该模型中，$\{t(e): e{an edge of}{\bf Z}^2 \}$是具有公共分布$F$的独立同分布族。我们用$T({\bf 0}, v)$表示从原点到$v$的通过时间，对于$v\in {\bf R}^2$和$B(t)=\{v\in {\bf R}^d: T({\bf 0}, v)\leq t\}.$。众所周知，如果$F(0) < p_c$，存在一个紧凑的形状${\bf B}_F\subset {\bf R}^2$，使得对于所有的$ε>0$，$t {\bf B}_F(1-ε) \subset {B(t)} \subset t{\bf B}_F(1+ε)$，最终概率为1。对于每个形状边界点$u$，我们用$κ^+(u)$和$κ^-(u)$表示其右曲率指数和左曲率指数。此外，对于每个矢量$u$，我们用$ξ(u)$表示横向波动指数。在本文中，我们可以证明，$ξ(u) \leq 1-\max\{κ^-(u)/2, κ^+(u)/2\}$的所有形状的边界点$u$。为了追求${\bf B}_F$上的曲率，我们考虑具有特殊分布的通过时间infsuppPDMATHABNC和$F(l)=p > \vec{p}_c$，其中$l$是正数，$\vec{p}_c$是定向渗流模型的临界点。利用该分布，已知在角度$0< θ_p^- < θ_p^+< 90^\…
The Brownian web is a two-dimensional black noise
布朗网是一个随机变量，由平面上每个时空点的布朗运动组成。它们是独立的，直到它们彼此碰撞，在这一点上它们合并。Tsirelson在他的论文“Scaling limit，Noise，Stability”中提到了这个模型，以及平面渗流，提出了二维黑噪声的存在。粗略地说，二维噪声是平面上的随机对象，其分布是平移不变的，并且其在不相交子集上的行为是独立的。黑色意味着对任意小的总面积的集合的恢复敏感。Tsirelson含蓄地问道：“布朗网络是一个二维的黑噪声吗？".我们给出了一个肯定的答案，这个问题，提供了第二个已知的例子后，临界平面渗流的缩放限制。
Clusters in middle-phase percolation on hyperbolic plane
我认为p-Bernoulli键渗流图的顶点传递平铺的双曲平面与有限的侧面（或，等价地说，传递，不可测，可平面图与一端），并在其上。已知（Benjamini和Schramm），在这样的图G中，我们有三个基本的渗流阶段，即：e. 0 < p_c（G）< p_u（G）< 1，其中p_c为临界概率，p_u为统一概率。我在中间阶段证明了这一点。S.所有无限簇的所有端点都在H^2的边界上有一点边界。这个结果与Lalley的一些结果相似。
Fluctuations around the diagonal in Bernoulli-Exponential first passage percolation
本文研究 Bernoulli-Exponential 首通渗流中团簇边界在对角线附近的涨落，证明其重标度单点涨落收敛于以渗流水平 s 为索引的新分布族，该族在 s=0 时为高斯分布，并随 s 增大趋于 Tracy--Widom 分布。
Multivariate spatial central limit theorems with applications to percolation and spatial graphs
假设$Z^d)$中的$X = (X_x, x$是i.i.d.对于某些可测量空间中的变量，$B_0$是$R^d$中的有界集合，并且对于$t > 1$，$H_t$是$tB_0$上的测量，该测量由$X$对$tB_0$中或附近的晶格位点的限制确定。我们证明了收敛到一个白噪声过程的随机测量$B_0$上给出的$t^{-d/2}(H_t(tA)-EH_t(tA))$的子集$A$的$B_0$，$t$变得大，受到$H$满足"稳定“条件（从而改变$X$在一个单一的网站$x$的影响），但没有假设的衰减率的相关性。我们还给出了一个多元中心极限定理的联合分布的两个或两个以上这样的措施$H_t$，并适应的结果的泊松和二项式点过程的基础上的措施。给出的应用包括白噪声限制的措施，计数集群的临界渗流，一个功能中心极限定理的经验过程中的最小生成树的随机点的边长，和中心极限定理的在线最近邻图。
Oded Schramm's contributions to noise sensitivity
本文综述了Oded Schramm在噪声灵敏度方面的主要贡献。我们将特别介绍他的各种作品集中在“谱分析“的临界渗流（更一般的布尔函数），他的工作的形状波动的第一通道渗流，最后他的贡献模型的动态渗流。
We consider critical percolation on the triangular lattice in a bounded simply connected domain with boundary conditions that force an interface between two prescribed boundary points. We say the interface forms a "near-loop" when it comes within one lattice spacing of itself. We define a new curve by erasing these near-loops as we traverse the interface. Our Monte Carlo simulations of this model lead us to conclude that the scaling limit of this loop-erased percolation interface is conformally invariant and has fractal dimension 4/3. However, it is not SLE_8/3. We also consider the process in…
A percolation formula
设$A$是单位圆盘$U$边界上的弧。我们证明了一个渐近公式的概率，有一个渗流集团$K$临界点渗流的三角形网格上的$U$相交$A$，使0包围$K\cup A$。
Noise Sensitivity of Boolean Functions and Applications to Percolation
结果表明，在产品概率空间中的一个大类的事件是高度敏感的噪声，在这个意义上说，具有高概率，随机误差的任意小的百分比的配置几乎没有预测事件是否发生。另一方面，加权多数函数是噪声稳定的。给出了噪声灵敏度和稳定性的几个充要条件。例如，考虑$n+1$乘$n$网格上的键渗流。配置是一个函数，它为每条边指定值0或1。令$ω$是一个随机配置，根据均匀测量选择。交叉是连接矩形左右两侧的路径，完全由$e$和$ω(e)=1$组成。通过对偶性，有一个交叉的概率是1/2。修复$ε\in(0,1)$。对于每个边$e$，令概率为$1-ε$的$ω'(e)=ω(e)$和概率为$ε$的$ω'(e)=1-ω(e)$独立于其他边。令$p(τ)$为在$ω$中具有交叉的概率，条件为$ω'=τ$。那么对于所有足够大的$n$，则为$P\{τ: |p(τ)-1/2|>ε\}<ε$。
Proofs of sharp phase transition and noise sensitivity in percolation have been significantly simplified by the use of randomized algorithms, via the OSSS inequality (proved by O'Donnell, Saks, Schramm and Servedio (2005)) and the Schramm-Steif inequality for the Fourier-Walsh coefficients of functions defined on the Boolean hypercube. In this article, we prove intrinsic versions of the OSSS and Schramm-Steif inequalities for functionals of a general Poisson process, and apply these new estimates to deduce sufficient conditions - expressed in terms of randomized stopping sets - yielding sharp …
On the scaling limits of planar percolation
我们证明了Tsirelson的猜想，即临界平面渗流的任何标度极限都是黑噪声。我们的定理适用于一些渗流模型，包括网站上的三角形网格渗流和任何键渗流的平方网格上的连续标度限制。我们还提出了一个自然的建设规模限制的平面渗流，更一般的任何离散的平面模型描述的连通性。
We develop an unsupervised, nonparametric, and scalable statistical learning method for detection of unknown objects in noisy images. The method uses results from percolation theory and random graph theory. We present an algorithm that allows to detect objects of unknown shapes and sizes in the presence of nonparametric noise of unknown level. The noise density is assumed to be unknown and can be very irregular. The algorithm has linear complexity and exponential accuracy and is appropriate for real-time systems. We prove strong consistency and scalability of our method in this setup with mini…
Finite range decompositions of Gaussian fields with applications to level-set percolation
本文改进Bauerschmidt的有限程分解，证明一大类离散与连续高斯场具备带最优衰减参数的多尺度白噪声分解，从而支撑水平集渗流相变锐性并回答相关问题。
Despite great progress in the study of critical percolation on $\mathbb{Z}^d$ for $d$ large, properties of critical clusters in high-dimensional fractional spaces and boxes remain poorly understood, unlike the situation in two dimensions. Closely related models such as critical branching random walk give natural conjectures for the value of the relevant high-dimensional critical exponents; see in particular the conjecture by Kozma-Nachmias that the probability that $0$ and $(n, n, n, \ldots)$ are connected within $[-n,n]^d$ scales as $n^{-2-2d}$. In this paper, we study the properties of criti…
Partially observed Boolean sequences and noise sensitivity
令$\mathcal{H}$表示$\{1,2,\ldots,n\}$的子集的集合，并将均匀分布在$[0,1]$上的独立随机变量分配给$n$元素。如果元素$p$的相应值最多为$p$，则将其判定为存在。在本文中，我们量化了$r$存在（$r>p$）元素集的观察对$p$存在元素集包含在$\mathcal{H}$中的概率的影响。在渗流的背景下，我们发现，这个问题是密切相关的近临界政权。因此，我们表明，对于每一个$r>1/2$，键渗流的子图的正方形晶格所给出的一组$r$目前的边缘几乎肯定是噪声敏感的临界，从而推广了由于Benjamini，Kalai和Schramm的结果。
Examples of nonpolygonal limit shapes in i.i.d. first-passage percolation and infinite coexistence in spatial growth models
我们构造了i.i.d.的边权重分布.讨论了极限形状不是多边形且极值点在边界上任意稠密的$\mathbb{Z}^2$上的首过渗流问题。因此，相关的Richardson型增长模型可以支持可数无穷多个不同物种的共存，并且感染图具有无穷多个端点。
On the density of 2D critical percolation gaskets and anchored clusters
本文借助镜像法严格证明了锚定于边界点的二维临界渗流簇重整化密度公式，并建立了体—边界连接概率及任意共形圆盘区域中渗流垫圈密度的标度极限。
Critical speeding-up in dynamical percolation
本文研究离散时间动力学渗流中原点簇大小的自相关时间，证明在二叉树和高维环面中，该时间在次临界区域与体积成线性关系，而在临界点处成严格次线性关系，从而严格确立了临界加速现象。
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