Noam Berger
We consider long-range percolation in dimension $d\geq 1$, where distinct sites $x$ and $y$ are connected with probability $p_{x,y}\in[0,1]$. Assuming that $p_{x,y}$ is translation invariant and that $p_{x,y}=\|x-y\|^{-s+o(1)}$ with $s>2d$, we show that the graph distance is at least linear with the Euclidean distance.
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The aim of this paper is to generalize the well-known asymptotic shape result for first-passage percolation on $\Zd$ to first-passage percolation on a random environment given by the infinite cluster of a supercritical Bernoulli percolation model. We prove the convergence of the renormalized set of wet points to a deterministic shape that does not depend on the random environment. As a special case of the previous result, we obtain an asymptotic shape theorem for the chemical distance in supercritical Bernoulli percolation. We also prove a flat edge result. Some various examples are also given…
We continue our study of the chemical (graph) distance inside large critical percolation clusters in dimension two. We prove new estimates, which involve the three-arm probability, for the point-to-surface and point-to-point distances. We show that the point-to-point distance in $\mathbb{Z}^2$ between two points in the same critical percolation cluster has infinite second moment. We also give quantitative versions of our previous results comparing the length of the shortest crossing to that of the lowest crossing of a box.
In this paper, we establish moderate deviations for the chemical distance in Bernoulli percolation. The chemical distance between two points is the length of the shortest open path between these two points. Thus, we study the size of random fluctuations around the mean value, and also the asymptotic behavior of this mean value. The estimates we obtain improve our knowledge of the convergence to the asymptotic shape. Our proofs rely on concentration inequalities proved by Boucheron, Lugosi and Massart, and also on the approximation theory of subadditive functions initiated by Alexander.
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…
The ellipses model is a continuum percolation process in which ellipses with random orientation and eccentricity are placed in the plane according to a Poisson point process. A parameter $α$ controls the tail distribution of the major axis' distribution and we focus on the regime $α\in (1,2)$ for which there exists a unique infinite cluster of ellipses and this cluster fulfills the so called highway property. We prove that the distance within this infinite cluster behaves asymptotically like the (unrestricted) Euclidean distance in the plane. We also show that the chemical distance between poi…
We consider two-dimensional critical bond percolation. Conditioned on the existence of an open circuit in an annulus, we show that the ratio of the expected size of the shortest open circuit to the expected size of the innermost circuit tends to zero as the side length of the annulus tends to infinity, the aspect ratio remaining fixed. The same proof yields a similar result for the lowest open crossing of a rectangle. In this last case, we answer a question of Kesten and Zhang by showing in addition that the ratio of the length of the shortest crossing to the length of the lowest tends to zero…
The chemical distance D(x,y) is the length of the shortest open path between two points x and y in an infinite Bernoulli percolation cluster. In this work, we study the asymptotic behaviour of this random metric, and we prove that, for an appropriate norm $μ$ depending on the dimension and the percolation parameter, the probability of the event \[\biggl\{0\leftrightarrow x,\frac{D(0,x)}{μ(x)}\notin (1-ε, 1+ε) \biggr\}\] exponentially decreases when $\|x\|_1$ tends to infinity. From this bound we also derive a large deviation inequality for the corresponding asymptotic shape result.
We provide the first nontrivial upper bound for the chemical distance exponent in two-dimensional critical percolation. Specifically, we prove that the expected length of the shortest horizontal crossing path of a box of side length $n$ in critical percolation on $\mathbb{Z}^2$ is bounded by $Cn^{2-δ}π_3(n)$, for some $δ>0$, where $π_3(n)$ is the "three-arm probability to distance $n$." This implies that the ratio of this length to the length of the lowest crossing is bounded by an inverse power of $n$ with high probability. In the case of site percolation on the triangular lattice, we obtain …
In this note, we describe some of the progress recently made on questions regarding the chemical distance in two-dimensional critical percolation by the author, J. Hanson, and P. Sosoe [6, 7]. It is expected that the distance between points in critical percolation clusters scales as $\|\cdot \|^{1+s}$, where $\|\cdot \|$ is the Euclidean distance and $s>0$. First, we review previous work of Aizenman-Burchard and Morrow-Zhang, which together establish a version of $0 < s \leq 1/3$. The main results of our work are in the direction of proving upper bounds on $s$, answering in [6] a question from…
In this paper we provide general conditions on a one parameter family of random infinite subsets of Z^d to contain a unique infinite connected component for which the chemical distances are comparable to the Euclidean distances, focusing primarily on models with long-range correlations. Our results are in the spirit of those by Antal and Pisztora proved for Bernoulli percolation. We also prove a shape theorem for balls in the chemical distance under such conditions. Our general statements give novel results about the structure of the infinite connected component of the vacant set of random int…
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 study the two most common types of percolation process on a sparse random graph with a given degree sequence. Namely, we examine first a bond percolation process where the edges of the graph are retained with probability p and afterwards we focus on site percolation where the vertices are retained with probability p. We establish critical values for p above which a giant component emerges in both cases. Moreover, we show that in fact these coincide. As a special case, our results apply to power law random graphs. We obtain rigorous proofs for formulas derived by several physicists for such …
We consider an i.i.d. supercritical bond percolation on Z^d , every edge is open with a probability p > p\_c (d), where p\_c (d) denotes the critical parameter for this percolation. We know that there exists almost surely a unique infinite open cluster C\_p [11]. We are interested in the regularity properties of the chemical distance for supercritical Bernoulli percolation. The chemical distance between two points x, y $\in$ C\_p corresponds to the length of the shortest path in C\_p joining the two points. The chemical distance between 0 and nx grows asymptotically like n$μ$\_p (x). We aim to…
Near-critical percolation with sparse reinforcements
该研究探讨稀疏随机强化对方格点阵近临界伯努利点渗流的影响，揭示强化点集几何结构对相变临界阈值的调控规律及临界态渗流的矩条件。
We derive an estimate for the distance, measured in lattice spacings, inside two-dimensional critical percolation clusters from the origin to the boundary of the box of side length $2n$, conditioned on the existence of an open connection. The estimate we obtain is the radial analogue of the one found in the work of Damron, Hanson, and Sosoe. In the present case, however, there is no lowest crossing in the box to compare to, so we construct a path $γ$ from the origin to distance $n$ that consists of "three-arm" points, and whose volume can thus be estimated by $O(n^2π_3(n))$. Here, $π_3(n)$ is …
Shadow and percolation III: chemical distance in continuous landscapes with correlations
研究了斜率场α的 excursion 集合的几何性质，证明在超临界水平下，两点在 excusion 集合中的化学距离与欧几里得距离成正比，扩展了 Antal Pisztora 定理的应用范围。
On the upper tail large deviation rate function for chemical distance in supercritical percolation
本文研究了超临界键渗流中化学距离的上尾大偏差问题，证明了当 $d\ge 3$ 且 $\varepsilon>0$ 足够小时速率函数的存在性，并揭示该大偏差事件由时空切点迫使测地线绕行引发。
We consider the (unoriented) long-range percolation on Z^d in dimensions d\ge1, where distinct sites x,y\in Z^d get connected with probability p_{xy}\in[0,1]. Assuming p_{xy}=|x-y|^{-s+o(1)} as |x-y|\to\infty, where s>0 and |\cdot| is a norm distance on Z^d, and supposing that the resulting random graph contains an infinite connected component C_{\infty}, we let D(x,y) be the graph distance between x and y measured on C_{\infty}. Our main result is that, for s\in(d,2d), D(x,y)=(\log|x-y|)^{Δ+o(1)},\qquad x,y\in C_{\infty}, |x-y|\to\infty, where Δ^{-1} is the binary logarithm of 2d/s and o(1) i…
We show quenched large deviations for the simple random walk on a certain class of percolations with long-range correlations. This class contains the supercritical Bernoulli percolations, the model considered by Drewitz, R'ath and Sapozhnikov and the random-cluster model up to the slab critical point. Our result is an extension of Kubota's result for the supercritical Bernoulli percolations. We also state a shape theorem for the chemical distance, which is an extension of Garet and Marchand's result for the supercritical Bernoulli percolations.
We show that random walk on the incipient infinite cluster (IIC) of two-dimensional critical percolation is subdiffusive in the chemical distance (i.e., in the intrinsic graph metric). Kesten (1986) famously showed that this is true for the Euclidean distance, but it is known that the chemical distance is typically asymptotically larger. More generally, we show that subdiffusivity in the chemical distance holds for stationary random graphs of polynomial volume growth, as long as there is a multi-scale way of covering the graph so that "deep patches" have "thin backbones". Our estimates are qua…
Subdiffusive concentration for the chemical distance in Bernoulli percolation
针对超临界伯努利渗流中图距离的扩散集中性，本文建立次扩散集中不等式改进结果，受首达渗流工作启发并结合有效半径等新工具，以一步重正化处理边重采样影响，方差次线性界为其推论。
We study level-set percolation for Gaussian free fields on metric graphs. In two dimensions, we give an upper bound on the chemical distance between the two boundaries of a macroscopic annulus. Our bound holds with high probability conditioned on connectivity and the bound is sharp up to a poly-logarithmic factor with an exponent of one-quarter. This substantially improves a previous result by Li and the first author. In three dimensions and higher, we provide rather sharp estimates of percolation probabilities in different regimes which altogether describe a sharp phase transition.
We consider first passage percolation on sparse random graphs with prescribed degree distributions and general independent and identically distributed edge weights assumed to have a density. Assuming that the degree distribution satisfies a uniform X^2\log{X}-condition, we analyze the asymptotic distribution for the minimal weight path between a pair of typical vertices, as well the number of edges on this path or hopcount. The hopcount satisfies a central limit theorem where the norming constants are expressible in terms of the parameters of an associated continuous-time branching process. Ce…
In this paper we introduce the two-sided level-set for the two-dimensional discrete Gaussian free field. Then we investigate the chemical distance for the two-sided level-set percolation. Our result shows that the chemical distance should have dimension strictly larger than $1$, which in turn stimulates some tempting questions about the two-sided level-set.
Limiting distribution of the chemical distance in high dimensional critical percolation
该研究确定了高维临界及近临界渗流中化学距离等度量的极限分布，为理解高维簇的度量空间结构奠定基础，其证明依赖于开簇的混合结果。
We consider instances of long-range percolation on $\mathbb Z^d$ and $\mathbb R^d$, where points at distance $r$ get connected by an edge with probability proportional to $r^{-s}$, for $s\in (d,2d)$, and study the asymptotic of the graph-theoretical (a.k.a. chemical) distance $D(x,y)$ between $x$ and $y$ in the limit as $|x-y|\to\infty$. For the model on $\mathbb Z^d$ we show that, in probability as $|x|\to\infty$, the distance $D(0,x)$ is squeezed between two positive multiples of $(\log r)^Δ$, where $Δ:=1/\log_2(1/γ)$ for $γ:=s/(2d)$. For the model on $\mathbb R^d$ we show that $D(0,xr)$ is,…
Nontrivial upper bound for chemical distance on the planar random cluster model
本文将二维 Bernoulli 渗流的化学距离上界扩展至簇权重 $1 \le q \le 4$ 的平面随机簇模型，并为该模型的强臂分离引理提供了完整证明。
Superconcentration and chaos in Bernoulli percolation
本文研究了伯努利渗流中化学距离的超浓度现象，通过动态有效半径和晶格动物理论量化了测地线的混沌行为。
We make the first steps towards generalizing the theory of stochastic block models, in the sparse regime, towards a model where the discrete community structure is replaced by an underlying geometry. We consider a geometric random graph over a homogeneous metric space where the probability of two vertices to be connected is an arbitrary function of the distance. We give sufficient conditions under which the locations can be recovered (up to an isomorphism of the space) in the sparse regime. Moreover, we define a geometric counterpart of the model of flow of information on trees, due to Mossel …
The scaling limit of random walk and the intrinsic metric on planar critical percolation
研究揭示临界渗流中随机游走在标度极限下趋近于CLE_6布朗运动，同时证明内在度量趋近于CLE_6度量，推导出化学距离指数、电阻指数和谱维数，并验证爱因斯坦关系。
Lipschitz continuity of the time constant for continuum percolation
研究布尔模型连续渗流中时间常数关于强度的Lipschitz连续性，通过改编Can、Nakajima和Nguyen的论证方法证明。
Quantitative indistinguishability and sparse and dense clusters in factor of IID percolations
本文研究了因子IIDs渗流中非超有限聚类的量化不可区分性，证明了稀疏性与薄密度的关系，并展示了非超有限聚类的强不可区分性等性质。
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。
Arithmetic oscillations of the chemical distance in long-range percolation on $\mathbb Z^d$
本文研究了 $\mathbb Z^d$ 上长程渗流中典型顶点间的化学距离，证明其渐近于 $ϕ_β(r)(\log r)^Δ$，并否定了 $ϕ_β$ 为常数的猜想，揭示了最短路径的算术刚性。
In this paper, we study geometric properties of the unique infinite cluster $Γ$ in a sufficiently supercritical Finitary Random Interlacements $\mathcal{FI}^{u,T}$ in $\mathbb{Z}^d, \ d\ge 3$. We prove that the chemical distance in $Γ$ is, with stretched exponentially high probability, of the same order as the Euclidean distance in $\mathbb{Z}^d$. This also implies a shape theorem parallel to those for Bernoulli percolation and random interlacements. We also prove local uniqueness of $\mathcal{FI}^{u,T}$, which says any two large clusters in $\mathcal{FI}^{u,T}$ "close to each other" will with…
Consider the problem of determining the maximal induced subgraph in a random $d$-regular graph such that its components remain bounded as the size of the graph becomes arbitrarily large. We show, for asymptotically large $d$, that any such induced subgraph has size density at most $2(\log d)/d$ with high probability. A matching lower bound is known for independent sets. We also prove the analogous result for sparse Erdős-Rényi graphs.
The contact process on Scale-Free Percolation
本文研究无标度渗流随机图上接触过程的灭绝时间，证明在 $β\in(2,3)$ 与 $β\geq3$ 情形下其均随体积 n 指数增长，并通过多尺度分析、化学距离与渗流论证处理对数修正。
Chemical distance for smooth Gaussian fields in higher dimension
本文研究高维光滑高斯场超临界游走集中无界簇的几何，引入化学距离并证明其高概率近似于欧几里得距离。
This paper discusses first passage percolation and flooding on large weighted sparse random graphs with two types of nodes: active and passive nodes. In mathematical physics passive nodes can be interpreted as closed gates where fluid flow or water cannot pass through and active nodes can be interpreted as open gates where water may keep flowing further. The model of this paper has many applications in real life, for example, information spreading, where passive nodes are interpreted as passive receivers who may read messages but do not respond to them. In the epidemic context passive nodes ma…
We consider bond percolation on random graphs with given degrees and bounded average degree. In particular, we consider the order of the largest component after the random deletion of the edges of such a random graph. We give a rough characterisation of those degree distributions for which bond percolation with high probability leaves a component of linear order, known usually as a giant component. We show that essentially the critical condition has to do with the tail of the degree distribution. Our proof makes use of recent technique introduced by Joos et al. [FOCS 2016, pp. 695--703], which…
In high dimensional percolation at parameter $p < p_c$, the one-arm probability $π_p(n)$ is known to decay exponentially on scale $(p_c - p)^{-1/2}$. We show the same statement for the ratio $π_p(n) / π_{p_c}(n)$, establishing a form of a hypothesis of scaling theory. As part of our study, we provide sharp estimates (with matching upper and lower bounds) for several quantities of interest at the critical probability $p_c$. These include the tail behavior of volumes of, and chemical distances within, spanning clusters, along with the scaling of the two-point function at "mesoscopic distance" fr…
Chemical distance for the half-orthant model
本文研究了半象限随机介质模型的图距离（化学距离），在特定参数条件下利用次可加遍历定理证明了极限定理，并提出了证明期望步数有限及形状非平凡的新颖论证。
In this paper, we establish moderate deviations for the chemical distance in Bernoulli percolation. The chemical distance between two points is the length of the shortest open path between these two points. Thus, we study the size of random fluctuations around the mean value, and also the asymptotic behavior of this mean value. The estimates we obtain improve our knowledge of the convergence to the asymptotic shape. Our proofs rely on concentration inequalities proved by Boucheron, Lugosi and Massart, and also on the approximation theory of subadditive functions initiated by Alexander.
We study the asymptotic growth of the diameter of a graph obtained by adding sparse "long" edges to a square box in $\Z^d$. We focus on the cases when an edge between $x$ and $y$ is added with probability decaying with the Euclidean distance as $|x-y|^{-s+o(1)}$ when $|x-y|\to\infty$. For $s\in(d,2d)$ we show that the graph diameter for the graph reduced to a box of side $L$ scales like $(\log L)^{Δ+o(1)}$ where $Δ^{-1}:=\log_2(2d/s)$. In particular, the diameter grows about as fast as the typical graph distance between two vertices at distance $L$. We also show that a ball of radius $r$ in th…
Phase transition for level-set percolation of the membrane model in dimensions $d \geq 5$
该论文研究了维度 $d \geq 5$ 的高斯膜模型level-set渗流的相变，证明了临界水平的存在与性质，并利用新的解耦不等式刻画了连通函数的衰减和无限簇中的化学距离。
Recursive Packing Bounds for Supercritical Disconnection in Bernoulli Site Percolation
本文研究了伯努利位点渗流中超临界断开概率的量化上界，通过递归装箱数提出新的方法，并在射线同质树上展示了该数的显式计算。
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