Lincoln Chayes, Robin Pemantle, Yuval Peres
We show that fractal (or "Mandelbrot") percolation in two dimensions produces a set containing no directed paths, when the set produced has zero area. This improves a similar result by the first author in the case of constant retention probabilities to the case of retention probabilities approaching 1.
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We consider two variations on the Mandelbrot fractal percolation model. In the k-fractal percolation model, the d-dimensional unit cube is divided in N^d equal subcubes, k of which are retained while the others are discarded. The procedure is then iterated inside the retained cubes at all smaller scales. We show that the (properly rescaled) percolation critical value of this model converges to the critical value of ordinary site percolation on a particular d-dimensional lattice as N tends to infinity. This is analogous to the result of Falconer and Grimmett that the critical value for Mandelbr…
We study the porosity properties of fractal percolation sets $E\subset\mathbb{R}^d$. Among other things, for all $0<\varepsilon<\tfrac12$, we obtain dimension bounds for the set of exceptional points where the upper porosity of $E$ is less than $\tfrac12-\varepsilon$, or the lower porosity is larger than $\varepsilon$. Our method works also for inhomogeneous fractal percolation and more general random sets whose offspring distribution gives rise to a Galton-Watson process.
There are various notions of dimension in fractal geometry to characterise (random and non-random) subsets of $\mathbb R^d$. In this expository text, we discuss their analogues for infinite subsets of $\mathbb Z^d$ and, more generally, for infinite graphs. We then apply these notions to critical percolation clusters, where the various dimensions have different values.
Minkowski sum of fractal percolation and random sets
本文证明分形渗流Minkowski和的命中概率可由容量刻画，并推广至一般随机集，如随机游走和临界分支随机游走的轨迹。
We relate various concepts of fractal dimension of the limiting set C in fractal percolation to the dimensions of the set consisting of connected components larger than one point and its complement in C (the "dust"). In two dimensions, we also show that the set consisting of connected components larger than one point is a.s. the union of non-trivial Hölder continuous curves, all with the same exponent. Finally, we give a short proof of the fact that in two dimensions, any curve in the limiting set must have Hausdorff dimension strictly larger than 1.
Quasisymmetric mappings on two variants of fractal percolation
研究两类分形渗流变体上的拟对称映射，证明在非灭绝条件下其豪斯多夫维数不能被相应拟对称映射降低。
Partially motivated by the desire to better understand the connectivity phase transition in fractal percolation, we introduce and study a class of continuum fractal percolation models in dimension d greater than or equal to 2. These include a scale invariant version of the classical (Poisson) Boolean model of stochastic geometry and (for d=2) the Brownian loop soup introduced by Lawler and Werner. The models lead to random fractal sets whose connectivity properties depend on a parameter lambda. In this paper we mainly study the transition between a phase where the random fractal sets are total…
A well studied family of random fractals called fractal percolation is discussed. We focus on the projections of fractal percolation on the plane. Our goal is to present stronger versions of the classical Marstrand theorem, valid for almost every realization of fractal percolation. The extensions go in three directions: {itemize} the statements work for all directions, not almost all, the statements are true for more general projections, for example radial projections onto a circle, in the case $\dim_H >1$, each projection has not only positive Lebesgue measure but also has nonempty interior. …
A range of first-passage percolation type models are believed to demonstrate the related properties of sublinear variance and superdiffusivity. We show that directed last-passage percolation with Gaussian vertex weights has a sublinear variance property. We also consider other vertex weight distributions. Corresponding results are obtained for the ground state of the `directed polymers in a random environment' model.
We show that there exists $0<α_0<1$ (depending on the parameters) such that the fractal percolation is almost surely purely $α$-unrectifiable for all $α>α_0$.
Fractal percolation has been introduced by Mandelbrot in 1974. We study the two-dimensional case, with integer subdivision index M and survival probability p. It is well known that there exists a non-trivial critical value p_c(M) such that a.s. the largest connected component in the limiting set K is a point for p<p_c(M) and with positive probability there is a connected component intersecting opposite sides of the unit square for p\geq p_c(M). For all M\geq 2, the value of p_c(M) is unknown. In this paper we present ideas to find lower and upper bounds, significantly sharper than those alread…
The largest slice of fractal percolation
本文确定了平面分形渗流包含 $k$ 个共线点的维数阈值，并研究了临界维数下最大线性切片的广义豪斯多夫测度大小。
We show that the number of maximal paths in directed last-passage percolation on the hypercubic lattice ${\mathbb Z}^d$ $(d\geq2)$ in which weights take finitely many values is typically exponentially large.
We prove a GUE central limit theorem for random variables with finite fourth moment. We apply this theorem to prove that the directed first and last passage percolation problems in thin rectangles exhibit universal fluctuations given by the Tracy-Widom law.
This paper presents a new, short proof of the computation of the upper tail large deviation rate function for the Brownian directed percolation model. Through a distributional equivalence between the last passage time in this model and the largest eigenvalue in a random matrix drawn from the Gaussian Unitary Ensemble, this provides a new proof a previously known result. The method leads to associated results for the stationary Brownian directed percolation model which have not been observed before.
In this paper we study the radial and orthogonal projections and the distance sets of the random Cantor sets $E\subset \mathbb{R}^2 $ which are called Mandelbrot percolation or percolation fractals. We prove that the following assertion holds almost surely: if the Hausdorff dimension of $E$ is greater than 1 then the orthogonal projection to \textbf{every} line, the radial projection with \textbf{every} center, and distance set from \textbf{every} point contain intervals.
We study dimensional properties of visible parts of fractal percolation in the plane. Provided that the dimension of the fractal percolation is at least 1, we show that, conditioned on non-extinction, almost surely all visible parts from lines are 1-dimensional. Furthermore, almost all of them have positive and finite Hausdorff measure. We also verify analogous results for visible parts from points. These results are motivated by an open problem on the dimensions of visible parts.
We prove that the free energy of directed polymer in Bernoulli environment converges to the growth rate for the number of open paths in super-critical oriented percolation as the temperature tends to zero. Our proof is based on rate of convergence results which hold uniformly in the temperature. We also prove that the convergence rate is locally uniform in the percolation parameter inside the super-critical phase, which implies that the growth rate depends continuously on the percolation parameter.
We consider directed first passage percolation on the integer lattice, with time constant $μ$ and passage time $a_{0n}$ from the origin to $(n,0,...,0)$. It is shown that under certain conditions on the passage time distribution, $Ea_{0n} - nμ= O(n^{1/2}(\log\log n)/\log n)$.
For directed Bernoulli last passage percolation with i.i.d.~weights on vertices over a $n\times n$ grid and for $n$ large enough, the geodesics are shown to be concentrated in a cylinder, centered on the main diagonal and of width of order $n^{(2κ+2)/(2κ+3)}\sqrt{\ln n}$, where $1\leκ<\infty$ is the curvature power of the shape function at $(1,1)$. The methodology of proof is robust enough to also apply to directed Bernoulli first passage site percolation, and further to longest common subsequences in random words.
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。
\emph{Fractal percolation} or \emph{Mandelbrot percolation} is one of the most well studied families of random fractals. In this paper we study some of the geometric measure theoretical properties (dimension of projections and structure of slices) of these random sets. Although random, the geometry of those sets is quite regular. Our results imply that, denoting by $E\subset \mathbb{R}^2$ a typical realization of the fractal percolation on the plane, {itemize} If $\dim_{\rm H}E<1$ then for \textbf{all}lines $\ell$ the orthogonal projection $E_\ell$ of $E$ to $\ell$ has the same Hausdorff dimen…
A law of the iterated logarithm is established for the last passage times of directed percolation on rectangles in the plane over exponential or geometric independent random variables, rescaled to converge to the Tracy-Widom distribution. The normalization is of loglog type, in contrast with the log normalization for the largest eigenvalue of a GUE matrix recently put forward by E. Paquette and O. Zeitouni. The proof relies on sharp tail bounds and superadditivity, close to the standard law of the iterated logarithm. A weaker result for the liminf is also discussed.
The sequence of random probability measures $ν_n$ that gives a path of length $n$, $\unsur{n}$ times the sum of the random weights collected along the paths, is shown to satisfy a large deviations principle with good rate function the Legendre transform of the free energy of the associated directed polymer in a random environment. Consequences on the asymptotics of the typical number of paths whose collected weight is above a fixed proportion are then drawn.
We study the asymptotic quantization error of order $r$ for Markov-type measures $μ$ on a class of ratio-specified graph directed fractals. We show that the quantization dimension of $μ$ exists and determine its exact value $s_{r}$ in terms of spectral radius of a related matrix. We prove that the $s_{r}$-dimensional lower quantization coefficient of $μ$ is always positive. Moreover, inspired by Mauldin-Williams's work on the Hausdorff measure of graph directed fractals, we establish a necessary and sufficient condition for the $s_{r}$-dimensional upper quantization coefficient of $μ$ to be fi…
In this note, we study the directed first passage percolation introduced in [F. Comets, R. Fukushima, S. Nakajima and N. Yoshida: Journal of Statistical Physics, 161-(3), 577-597 (2015)]. It is proved that the shortest path from the origin to the $n$-th hyperplane makes a jump larger than a small power of $\log n$. Some numerical results are also provided, which indicates that the maximal jump size is much larger in a certain parameter region.
For first passage percolation (FPP) on integer lattice with i.i.d. passage time distributions, in order to show existence of semi-infinite geodesics along a fixed direction, one requires unproven assumptions on the limiting shape. We consider FPP on two-dimensional integer lattice with i.i.d. passage times distributed as Durrett-Liggett class of measures. For this model, we show that along any direction in a deterministic angular sector (known as percolation cone), starting from every lattice point there exists an infinite geodesic along that direction and such directed geodesics coalesce almo…
Following the recent investigations of Baik and Suidan in \cite{baik2005gcl} and Bodineau and Martin in \cite{bodineau2005upl}, we prove large deviation properties for a last-passage percolation model in $\mathbb{Z}^{2}_{+}$ whose paths are close to the axis. The results are mainly obtained when the random weights are Gaussian or have a finite moment-generating function and rely, as in \cite{baik2005gcl} and \cite{bodineau2005upl}, on an embedding in Brownian paths and the KMT approximation. The study of the subexponential case completes the exposition.
We consider a directed random walk on the backbone of the infinite cluster generated by supercritical oriented percolation, or equivalently the space-time embedding of the ``ancestral lineage'' of an individual in the stationary discrete-time contact process. We prove a law of large numbers and an annealed central limit theorem (i.e., averaged over the realisations of the cluster) using a regeneration approach. Furthermore, we obtain a quenched central limit theorem (i.e.\ for almost any realisation of the cluster) via an analysis of joint renewals of two independent walks on the same cluster.
Geometric functionals of fractal percolation
本文研究了分形渗流的几何函数，通过引入极限几何函数来确定渗流阈值。
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。
Let F1 and F2 be independent copies of correlated fractal percolation, with Hausdorff dimensions dimH(F1) and dimH(F2). Consider the following question: does dimH(F1)+dimH(F2)>1 imply that their algebraic difference F1-F2 will contain an interval? The well known Palis conjecture states that `generically' this should be true. Recent work by Kuijvenhoven and the first author (arXiv:0811.0525) on random Cantor sets can not answer this question as their condition on the joint survival distributions of the generating process is not satisfied by correlated fractal percolation. We develop a new condi…
We consider fractal percolation (or Mandelbrot percolation) which is one of the most well studied example of random Cantor sets. Rams and the first author studied the projections (orthogonal, radial and co-radial) of fractal percolation sets on the plane. We extend their results to higher dimension.
We study site and bond percolation on directed simple random graphs with a given degree distribution and derive the expressions for the critical value of the percolation probability above which the giant strongly connected component emerges and the fraction of vertices in this component.
We introduce a technique that uses projection properties of fractal percolation to establish dimension conservation results for sections of deterministic self-similar sets. For example, let $K$ be a self-similar subset of $\mathbb{R}^2$ with Hausdorff dimension $\dim_H K >1$ such that the rotational components of the underlying similarities generate the full rotation group. Then for all $ε>0$, writing $π_θ$ for projection onto the line $L_θ$ in direction $θ$, the Hausdorff dimensions of the sections satisfy $\dim_H (K\cap π_θ^{-1}x)> \dim_H K - 1 - ε$ for a set of $x \in L_θ$ of positive Lebes…
Recent Progress on Fractal Percolation
本文综述了分形渗流过程的近期研究进展，重点探讨了极限集的维数理论及超临界状态下非平凡连通分量的几何性质，并揭示了其Assouad谱和中间维数恒等于盒维数的结论。
Purely unrectifiable sets, fractal percolation and graphs of functions
本文综述了作者关于不可整流性的研究成果，包括Takagi、Weierstrass-Cellerier等连续函数图形上的纯不可整流1-集，以及分形渗流几乎必然纯α-不可整流的条件。
Directed fractal percolation and non-Lipschitz variants
本文研究分形渗流中的最后通行渗流量化问题，开发多尺度框架证明通行时间与线性增长有几乎多项式差距，提出不可矩形定向渗流模型以构造符合要求的定向路径。
Visible parts and projections of the affine fractal percolation
该研究针对仿射分形渗流，明确了其可见部分在不同方向的维数规律及主收缩方向的特殊公式，还得出了其正交投影的豪斯多夫维数性质与区间包含结论。
Almost sure convergence and second moments of geometric functionals of fractal percolation
本文研究了分形渗流构造步骤中几何函数的几乎处处收敛与二阶矩，揭示了内禀体积的极限变量分解及其期望和协方差结构，并给出了收敛速率。
We investigate extended processes given by last-passage times in directed models defined using exponential variables with decaying mean. In certain cases we find the universal Airy process, but other cases lead to non-universal and trivial extended processes.
We introduce a generalization of the extended Airy kernel with two sets of real parameters. We show that this kernel arises in the edge scaling limit of correlation kernels of determinantal processes related to a directed percolation model and to an ensemble of random matrices.
The fractal dimension of minimal spanning trees on percolation clusters is estimated for dimensions $d$ up to $d=5$. A robust analysis technique is developed for correlated data, as seen in such trees. This should be a robust method suitable for analyzing a wide array of randomly generated fractal structures. The trees analyzed using these techniques are built using a combination of Prim's and Kruskal's algorithms for finding minimal spanning trees. This combination reduces memory usage and allows for simulation of larger systems than would otherwise be possible. The path length fractal dimens…
We study Mandelbrot's percolation process in dimension $d \geq 2$. The process generates random fractal sets by an iterative procedure which starts by dividing the unit cube $[0,1]^d$ in $N^d$ subcubes, and independently retaining or discarding each subcube with probability $p$ or $1-p$ respectively. This step is then repeated within the retained subcubes at all scales. As $p$ is varied, there is a percolation phase transition in terms of paths for all $d \geq 2$, and in terms of $(d-1)$-dimensional "sheets" for all $d \geq 3$. For any $d \geq 2$, we consider the random fractal set produced at…
Techniques of `dynamic renormalization', developed earlier for undirected percolation and the contact model, are adapted to the setting of directed percolation, thereby obtaining solutions of several problems for directed percolation on $Z^d$ where $d \ge 2$. The first new result is a type of uniqueness theorem: for every pair $x$ and $y$ of vertices which lie in infinite open paths, there exists almost surely a third vertex $z$ which is joined to infinity and which is attainable from $x$ and $y$ along directed open paths. Secondly, it is proved that a random walk on an infinite directed clust…
One-point distribution of the geodesic in directed last passage percolation
针对具有独立同分布指数权重的定向最后通过渗流，本文给出测地线位置与最后通过时间的显式联合一点分布，并证明其在 KPZ 标度下的极限。
In the previous article (S. Matsutani and Y. Shimosako and Y. Wang, Physica A \bf{391} (2012) 5802-5809) we numerically investigated an electric potential problem with high contrast local conductivities ($γ_0$ and $γ_1$, $0<γ_0 \ll γ_1$) for a two-dimensional continuum percolation model (CPM). As numerical results, we showed there that the equipotential curves exhibit the fractal structure around the threshold $p_c$ and gave an approximated curve representing a relation between the homogenized conductivity and the volume fraction $p$ over $[p_c,1]$. In this article, using the duality of the co…
Suppose each site independently and randomly chooses some sites around it, and it is weakly (strongly) connected with them (if there choose each other). What is the probability that the weak (strong) connected cluster is infinite? We investigate a percolation model for this problem, which is a generalization of site percolation. We give a relation between the probability of the number of chosen sites around a site and the size of clusters. We also see the expected number of infinite clusters, and the exponential tail decay of the radius and the size of a cluster.
Previous studies of connectivity in wireless networks have focused on undirected geometric graphs. More sophisticated models such as Signal-to-Interference-and-Noise-Ratio (SINR) model, however, usually leads to directed graphs. In this paper, we study percolation processes in wireless networks modelled by directed SINR graphs. We first investigate interference-free networks, where we define four types of phase transitions and show that they take place at the same time. By coupling the directed SINR graph with two other undirected SINR graphs, we further obtain analytical upper and lower bound…
In directed last passage site percolation with i.i.d.~random weights with finite support over a $n\times\lfloor n^α\rfloor$ grid, we prove that for $n$ large enough, the order of the $r$-th central moment, $1\le r<+\infty$, of the last passage time is lower bounded by $n^{r(1-α)/2}$, $0<α<1/3$.
We consider a directed percolation process on an ${\cal M}$ x ${\cal N}$ rectangular lattice whose vertical edges are directed upward with an occupation probability y and horizontal edges directed toward the right with occupation probabilities x and 1 in alternate rows. We deduce a closed-form expression for the percolation probability P(x,y), the probability that one or more directed paths connect the lower-left and upper-right corner sites of the lattice. It is shown that P(x,y) is critical in the aspect ratio $a = {\cal M}/{\cal N}$ at a value $a_c =[1-y^2-x(1-y)^2]/2y^2$ where P(x,y) is di…
In this paper, we obtain optimal uniform lower tail estimates for the probability distribution of the properly scaled length of the longest up/right path of the last passage site percolation model considered by Johansson in [12]. The estimates are used to prove a lower tail moderate deviation result for the model. The estimates also imply the convergence of moments, and also provide a verification of the universal scaling law relating the longitudinal and the transversal fluctuations of the model.
We study the time constant $C(ν)$ of last passage percolation on the complete directed acyclic graph on the set of non-negative integers, where edges have i.i.d. weights with distribution $ν$ with support included in $\{-\infty\}\cup\mathbb{R}$. We show that $ν\mapsto C(ν)$ is strictly increasing in $ν$. We also prove that $C(ν)$ is continuous in $ν$ for a large set of measures $ν$. Furthermore, when $ν$ is purely atomic, we show that $C(ν)$ is analytic with respect to the weights of the atoms. In the special case of two positive atoms, it is an explicit rational function of these weights.
Directed last passage percolation with discontinuous weights
研究证明了具有不连续宏观不均匀性的定向最后通过渗透模型在连续极限下对应于求解粘性Hamilton-Jacobi方程，该方程与总不对称简单排斥过程的水动力极限守恒律密切相关，并提出了一种基于动态规划的算法用于寻找最大定向路径的渐近形状。
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