The Diameter of Long-Range Percolation Clusters on Finite Cycles

Voronoi 渗流：拓扑稳定性与巨环

Voronoi Percolation: Topological Stability and Giant Cycles

研究高维 Voronoi 渗流的拓扑稳定性，通过微增 p 实现离散化并保持拓扑性质，推广 Bollobás-Riordan 定理，证明 2i 维环面上 i 维巨环涌现的尖锐相变。

Diameters in supercritical random graphs via first passage percolation

We study the diameter of $C_1$, the largest component of the Erdős-Rényi random graph $G(n,p)$ in the emerging supercritical phase, i.e., for $p = \frac{1+ε}n$ where $ε^3 n \to \infty$ and $ε=o(1)$. This parameter was extensively studied for fixed $ε> 0$, yet results for $ε=o(1)$ outside the critical window were only obtained very recently. Prior to this work, Riordan and Wormald gave precise estimates on the diameter, however these did not cover the entire supercritical regime (namely, when $ε^3 n\to\infty$ arbitrarily slowly). Łuczak and Seierstad estimated its order throughout this regime, …

Long Range Percolation Mixing Time

We provide an estimate, sharp up to poly-logarithmic factors, of the asymptotically almost sure mixing time of the graph created by long-range percolation on the cycle of length N (Z/NZ). While it is known that the almost sure diameter drops from linear to poly-logarithmic as the exponent s decreases below 2, the almost sure mixing time drops from N^2 only to N^(s-1) (up to poly-logarithmic factors).

Two-dimensional volume-frozen percolation: exceptional scales

We study a percolation model on the square lattice, where clusters "freeze" (stop growing) as soon as their volume (i.e. the number of sites they contain) gets larger than N, the parameter of the model. A model where clusters freeze when they reach diameter at least N was studied in earlier papers. Using volume as a way to measure the size of a cluster - instead of diameter - leads, for large N, to a quite different behavior (contrary to what happens on the binary tree, where the volume model and the diameter model are "asymptotically the same"). In particular, we show the existence of a seque…

A sharp leading order asymptotic of the diameter of a long range percolation graph

Many real-world networks exhibit the so-called small-world phenomenon: their typical distances are much smaller than their sizes. One mathematical model for this phenomenon is a long-range percolation graph on a $d$-dimensional box $\{0, 1, \cdots, N\}^d$, in which edges are independently added between far-away sites with probability falling off as a power of the Euclidean distance. A natural question is how the resulting diameter of the box of size $N$, measured in graph-theoretical distance, scales with $N$. This question has been intensely studied in the past and the answer depends on the e…

Boundary rules and breaking of self-organized criticality in 2D frozen percolation

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…

One-arm exponent for critical 2D percolation

The probability that the cluster of the origin in critical site percolation on the triangular grid has diameter larger than $R$ is proved to decay like $R^{-5/48}$ as $R\to\infty$.

Corner percolation on $\mathbb{Z}^2$ and the square root of 17

We consider a four-vertex model introduced by Bálint Tóth: a dependent bond percolation model on $\mathbb{Z}^2$ in which every edge is present with probability 1/2 and each vertex has exactly two incident edges, perpendicular to each other. We prove that all components are finite cycles almost surely, but the expected diameter of the cycle containing the origin is infinite. Moreover, we derive the following critical exponents: the tail probability $\mathbb{P}$(diameter of the cycle of the origin $>$$n$) $\approx$ $n^{-γ}$ and the expectation $\mathbb{E}$(length of a typical cycle with diameter…

A note about critical percolation on finite graphs

In this note we study the geometry of the largest component C_1 of critical percolation on a finite graph G which satisfies the finite triangle condition, defined by Borgs et al. There it is shown that this component is of size n^{2/3}, and here we show that its diameter is n^{1/3} and that the simple random walk takes n steps to mix on it. Our results apply to critical percolation on several high-dimensional finite graphs such as the finite torus Z_n^d (with d large and n tending to infinity) and the Hamming cube {0,1}^n.

Slightly subcritical hypercube percolation

We study bond percolation on the hypercube $\{0,1\}^m$ in the slightly subcritical regime where $p = p_c (1-\varepsilon_m)$ and $\varepsilon_m = o(1)$ but $\varepsilon_m \gg 2^{-m/3}$ and study the clusters of largest volume and diameter. We establish that with high probability the largest component has cardinality $Θ\left(\varepsilon_m^{-2} \log(\varepsilon_m^3 2^m)\right)$, that the maximal diameter of all clusters is $(1+o(1)) \varepsilon_m^{-1} \log(\varepsilon_m^3 2^m)$, and that the maximal mixing time of all clusters is $Θ\left(\varepsilon_m^{-3} \log^2(\varepsilon_m^3 2^m)\right)$. The…

Cycle structure of percolation on high-dimensional tori

In the past years, many properties of the largest connected components of critical percolation on the high-dimensional torus, such as their sizes and diameter, have been established. The order of magnitude of these quantities equals the one for percolation on the complete graph or Erdos-Renyi random graph, raising the question whether the scaling limits of the largest connected components, as identified by Aldous (1997), are also equal. In this paper, we investigate the cycle structure of the largest critical components for high-dimensional percolation on the torus (Z/rZ)^d. While percolation …

环面上的同调渗流：plaquettes 与 permutohedra

Homological percolation on a torus: plaquettes and permutohedra

本文构建环面上的高维同调渗流模型，证明巨型同调循环存在尖锐相变，并在中间维数得到临界阈值 $p_c=1/2$，推广 Kesten 的经典结果。

Revision of upper estimate of percolation threshold on square lattice

The more exact upper estimate of the percolation threshold for the {\it site problem} on the quadratic lattice ${\Bbb Z}^2$ have been found on the basis of the cluster decomposition. It is done by the number estimate of cycles on ${\Bbb Z}^2$ which maybe external boundaries of finite clusters

关于临界高维渗流中环的研究

On Loops in critical high-dimensional percolation

研究揭示高维临界伯努利渗流中存在自回避开放环，其尺度与盒子边长相当，且这些环的标度极限非平凡，与Aizenman三十年前指出的典型渗流簇不同。

Bootstrap percolation on products of cycles and complete graphs

Bootstrap percolation on a graph iteratively enlarges a set of occupied sites by adjoining points with at least $θ$ occupied neighbors. The initially occupied set is random, given by a uniform product measure, and we say that spanning occurs if every point eventually becomes occupied. The main question concerns the critical probability, that is, the minimal initial density that makes spanning likely. The graphs we consider are products of cycles of $m$ points and complete graphs of $n$ points. The major part of the paper focuses on the case when two factors are complete graphs and one factor i…

Duality in percolation via outermost boundaries I: Bond Percolation

Tile (\mathbb{R}^2\) into disjoint unit squares (\{S_k\}_{k \geq 0}\) with the origin being the centre of \(S_0\) and say that (S_i\) and (S_j\) are star adjacent if they share a corner and plus adjacent if they share an edge. Every square is either vacant or occupied. Outermost boundaries of finite star and plus connected components frequently arise in the context of contour analysis in percolation and random graphs. In this paper, we derive the outermost boundaries for finite star and plus connected components using a piecewise cycle merging algorithm. For plus connected components, the oute…

Heat Kernel Upper Bounds on Long Range Percolation Clusters

In this paper, we derive upper bounds for the heat kernel of the simple random walk on the infinite cluster of a supercritical long range percolation process. For any $d \geq 1$ and for any exponent $s \in (d, (d+2) \wedge 2d)$ giving the rate of decay of the percolation process, we show that the return probability decays like $t^{-\ffrac{d}{s-d}}$ up to logarithmic corrections, where $t$ denotes the time the walk is run. Moreover, our methods also yield generalized bounds on the spectral gap of the dynamics and on the diameter of the largest component in a box. Besides its intrinsic interest,…

Phase transitions for the Boolean model of continuum percolation for Cox point processes

We consider the Boolean model with random radii based on Cox point processes. Under a condition of stabilization for the random environment, we establish existence and non-existence of subcritical regimes for the size of the cluster at the origin in terms of volume, diameter and number of points. Further, we prove uniqueness of the infinite cluster for sufficiently connected environments.

Frozen percolation in two dimensions

Aldous introduced a modification of the bond percolation process on the binary tree where clusters stop growing (freeze) as soon as they become infinite. We investigate the site version of this process on the triangular lattice where clusters freeze as soon as they reach L infinity diameter at least N for some parameter N. We show that, informally speaking, in the limit N goes to infinity, the clusters only freeze in the critical window of site percolation on the triangular lattice. Hence the fraction of vertices that eventually (i. e. at time 1) are in a frozen cluster tends to 0 as N goes to…

Subcritical Connectivity and Some Exact Tail Exponents in High Dimensional Percolation

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…

有限图上的渗流

Percolation on Finite Graphs

本讲义基于特拉维夫大学研究生课程，系统介绍有限图上的渗流理论，涵盖随机图相变、巨分量、长路径、连通性与完美匹配阈值及随机超立方体中的相关结果。

Distances in critical long range percolation

We study the long range percolation model on $\mathbb{Z}$ where sites $i$ and $j$ are connected with probability $β|i-j|^{-s}$. Graph distances are now well understood for all exponents $s$ except in the case $s=2$ where the model exhibits non-trivial self-similar scaling. Establishing a conjecture of Benjamini and Berger \cite{BenBer:01}, we prove that the typical distance from site 0 to $n$ grows as a power law $n^{θ(β)}$ up to a multiplicative constant for some exponent $0<θ(β)<1$ as does the diameter of the graph on a box of length $n$.

Sharpness of the phase transition and lower bounds for the critical intensity in continuum percolation on $\mathbb{R}^d$

We consider the Boolean model $Z$ on $\mathbb{R}^d$ with random compact grains, i.e. $Z := \bigcup_{i \in \mathbb{N}} (X_i + Z_i)$ where $η_t := \{X_1, X_2, \dots\}$ is a Poisson point process of intensity $t$ and $(Z_1, Z_2, \dots)$ is an i.i.d. sequence of compact grains (not necessarily balls). We will show, that the volume and diameter of the cluster of a typical grain in $Z$ have an exponential tail if the diameter of the typical grain is a.s. bounded by some constant. To achieve this we adapt the arguments of \cite{duminil2015newproof} and apply a new construction of the cluster of the t…

Bounds for the annealed return probability on large finite percolation clusters

Bounds for the expected return probability of the delayed random walk on finite clusters of an invariant percolation on transitive unimodular graphs are derived. They are particularly suited for the case of critical Bernoulli percolation and the associated heavy-tailed cluster size distributions. The upper bound relies on the fact that cartesian products of finite graphs with cycles of a certain minimal size are Hamiltonian. For critical Bernoulli bond percolation on the homogeneous tree this bound is sharp. The asymptotic type of the expected return probability for large times t in this case …

Two-dimensional volume-frozen percolation: deconcentration and prevalence of mesoscopic clusters

Frozen percolation on the binary tree was introduced by Aldous around fifteen years ago, inspired by sol-gel transitions. We investigate a version of the model on the triangular lattice, where connected components stop growing ("freeze") as soon as they contain at least $N$ vertices, for some parameter $N \geq 1$. This process has a substantially different behavior from the diameter-frozen process, studied in previous works: in particular, we show that many (more and more as $N \to \infty$) frozen clusters surrounding the origin appear successively, each new cluster having a diameter much smal…

Critical random graphs: Diameter and mixing time

Let $\mathcal{C}_1$ denote the largest connected component of the critical Erdős--Rényi random graph $G(n,{\frac{1}{n}})$. We show that, typically, the diameter of $\mathcal{C}_1$ is of order $n^{1/3}$ and the mixing time of the lazy simple random walk on $\mathcal{C}_1$ is of order $n$. The latter answers a question of Benjamini, Kozma and Wormald. These results extend to clusters of size $n^{2/3}$ of $p$-bond percolation on any $d$-regular $n$-vertex graph where such clusters exist, provided that $p(d-1)\le1+O(n^{-1/3})$.

Percolation in Random Graphs: A Finite Approach

We propose an approach to calculate the critical percolation threshold for finite-sized Erdos-Renyi digraphs using minimal Hamiltonian cycles. We obtain an analytically exact result, valid non-asymptotically for all graph sizes, which scales in accordance with results obtained for infinite random graphs using the emergence of a giant connected component as marking the percolation transition. Our approach is general and can be applied to all graph models for which an algebraic formulation of the adjacency matrix is available.

A modified bootstrap percolation on a random graph coupled with a lattice

In this paper a random graph model $G_{\mathbb{Z}^2_N,p_d}$ is introduced, which is a combination of fixed torus grid edges in $(\mathbb{Z}/N \mathbb{Z})^2$ and some additional random ones. The random edges are called long, and the probability of having a long edge between vertices $u,v\in(\mathbb{Z}/N \mathbb{Z})^2$ with graph distance $d$ on the torus grid is $p_d=c/Nd$, where $c$ is some constant. We show that, {\em whp}, the diameter $D(G_{\mathbb{Z}^2_N,p_d})=Θ(\log N)$. Moreover, we consider non-monotonous bootstrap percolation on $G_{\mathbb{Z}^2_N,p_d}$. We prove the presence of phase …

On chemical distances and shape theorems in percolation models with long-range correlations

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…

维数至少为三的超临界边渗流中有限点到平面连接的精确渐近性

Sharp asymptotics for finite point-to-plane connections in supercritical bond percolation in dimension at least three

研究 $d\geq3$ 超临界边渗流中有限开簇到达特定平面的概率，通过证明长簇中更新点丰富且更新块长度尾部快速衰减，导出了轴向与一般方向下的精确渐近公式。

Transience and recurrence of random walks on percolation clusters in an ultrametric space

We study existence of percolation in the hierarchical group of order $N$, which is an ultrametric space, and transience and recurrence of random walks on the percolation clusters. The connection probability on the hierarchical group for two points separated by distance $k$ is of the form $c_k/N^{k(1+δ)}, δ>-1$, with $c_k=C_0+C_1\log k+C_2k^α$, non-negative constants $C_0, C_1, C_2$, and $α>0$. Percolation was proved in Dawson and Gorostiza (2013) for $δ<1$, and for the critical case, $δ=1,C_2>0$, with $α>2$. In this paper we improve the result for the critical case by showing percolation for $…

扩展周期图上首通渗的相变与波动结果

Phase Transition and Fluctuation Results for First-Passage Percolation on Spread-Out Cycle Graphs

该研究分析了ℓ-扩展一维周期图上首通渗的两种几何区域（介观、宏观）的渐近行为，确定了临界连通性阈值，实现了从高斯波动到极值波动的范围插值。

The Largest Cluster in Subcritical Percolation

The statistical behavior of the size (or mass) of the largest cluster in subcritical percolation on a finite lattice of size $N$ is investigated (below the upper critical dimension, presumably $d_c=6$). It is argued that as $N \to \infty$ the cumulative distribution function converges to the Fisher-Tippett (or Gumbel) distribution $e^{-e^{-z}}$ in a certain weak sense (when suitably normalized). The mean grows like $s_ξ^* \log N$, where $s_ξ^*(p)$ is a ``crossover size''. The standard deviation is bounded near $s_ξ^* π/\sqrt{6}$ with persistent fluctuations due to discreteness. These predictio…

A percolation process on the square lattice where large finite clusters are frozen

Aldous constructed a growth process for the binary tree where clusters freeze as soon as they become infinite. It was pointed out by Benjamini and Schramm that such a process does not exist for the square lattice. This motivated us to investigate the modified process on the square lattice, where clusters freeze as soon as they have diameter larger than or equal to N, the parameter of the model. The non-existence result, mentioned above, raises the question if the N-parameter model shows some 'anomalous' behaviour as N tends to infinity. For instance, if one looks at the cluster of a given vert…

Bidirectional bond percolation model for the spread of information in financial markets

Information is a key component in determining the price of an asset in financial markets, and the main objective of this paper is to study the spread of information in this context. The network of interactions in financial markets is modeled using a Galton-Watson tree where vertices represent the traders and where two traders are connected by an edge if one of the two traders sells the asset to the other trader. The information starts from a given vertex and spreads through the edges of the graph going independently from seller to buyer with probability $p$ and from buyer to seller with probab…

环面上的长程首达渗流

Long-range first-passage percolation on the torus

研究环面上完全图的长程首达渗流，在α∈[0,d)区间内，给出典型距离、洪泛时间和直径的1,2,3型结果，推广了Janson的完全图情形并为Chatterjee-Dey的定性结果提供定量视角。

布朗运动环汤中高维空间的强度倍增

Intensity doubling for Brownian loop-soups in high dimensions

研究高维空间中布朗运动环汤的强度倍增现象，通过分解环簇发现两种分布相同的家族，揭示了大规模循环在标度极限下的行为。

关于 Poisson 连续渗流的正态逼近速率

On the rate of normal approximation for Poisson continuum percolation

本文研究了渗流Poisson过程最大簇点数的正态逼近收敛速率，通过利用第二大簇较短依赖范围限制长程依赖，结合Chen等(2021)结果获得Berry-Esseen型界限并估计差距。

魔方群直径的图论估计

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。

热力学 regime 下通过随机几何图检测一阶同调

Detection of first homology via random geometric graphs in the thermodynamic regime

该研究针对紧致黎曼流形上的随机几何图，提出仅需 O(n) 条边的方法，在热力学超临界 regime 下可高概率推断其一阶同调群，确定了正确尺度并证明结果最优。