相关论文: Neighboring clusters in Bernoulli percolation
We prove that for supercritical percolation on every infinite transitive graph, the probability that the origin belongs to a finite cluster of size at least $n$ decays exponentially in $\Phi(n)$, where $\Phi$ is the isoperimetric function…
Several results are presented for site percolation on quasi-transitive, planar graphs $G$ with one end, when properly embedded in either the Euclidean or hyperbolic plane. If $(G_1,G_2)$ is a matching pair derived from some quasi-transitive…
This paper explores the link between the ergodicity of the clus-ter equivalence relation restricted to its infinite locus and the indis-tinguishability of infinite clusters. It is an important element of the dictionary connecting orbit…
We show that adding epsilon-Bernoulli percolation to an everywhere percolating subgraph of Z^2 results in a graph which has large scale geometry similar to that of supercritical Bernoulli percolation, in various specific senses. We…
We prove quasi-multiplicativity for critical level-sets of Gaussian free fields (GFF) on the metric graphs $\widetilde{\mathbb{Z}}^d$ ($d\ge 3$). Specifically, we study the probability of connecting two general sets located on opposite…
Given a weighted graph, we introduce a partition of its vertex set such that the distance between any two clusters is bounded from below by a power of the minimum weight of both clusters. This partition is obtained by recursively merging…
We introduce a formalism for computing bond percolation properties of a class of correlated and clustered random graphs. This class of graphs is a generalization of the Configuration Model where nodes of different types are connected via…
Consider percolation on $T\times \mathbb{Z}^d$, the product of a regular tree of degree $k\geq 3$ with the hypercubic lattice $\mathbb{Z}^d$. It is known that this graph has $0<p_c<p_u<1$, so that there are non-trivial regimes in which…
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…
A class of countable infinite graphs with unbounded vertex degree is considered. In these graphs, the vertices of large degree `repel' each other, which means that the path distance between two such vertices cannot be smaller than a certain…
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…
This is a study of percolation in the hyperbolic plane and on regular tilings in the hyperbolic plane. The processes discussed include Bernoulli site and bond percolation on planar hyperbolic graphs, invariant dependent percolations on such…
Let $(G_n) = \left((V_n,E_n)\right)$ be a sequence of finite connected vertex-transitive graphs with uniformly bounded vertex degrees such that $\lvert V_n \rvert \to \infty$ as $n \to \infty$. We say that percolation on $G_n$ has a sharp…
Breaking of equivalence between the microcanonical ensemble and the canonical ensemble, describing a large system subject to hard and soft constraints, respectively, was recently shown to occur in large random graphs. Hard constraints must…
We consider infinite connected quasi-transitive locally finite graphs and show that every such graph with more than one end is a tree amalgamation of two other such graphs. This can be seen as a graph-theoretical version of Stallings'…
A pair of complementary algorithms are presented. One of the pair is a fast method for connecting graphs with an edge. The other is a fast method for removing edges from a graph. Both algorithms employ the same tree based graph…
We present a novel hierarchical graph clustering algorithm inspired by modularity-based clustering techniques. The algorithm is agglomerative and based on a simple distance between clusters induced by the probability of sampling node pairs.…
We study the problem of coexistence in a two-type competition model governed by first-passage percolation on $\Zd$ or on the infinite cluster in Bernoulli percolation. Actually, we prove for a large class of ergodic stationary passage times…
This paper focuses on the asymptotic behaviors of the length of the largest 1-cluster in a finite iid Bernoulli sequence. We first reveal a critical phenomenon on the length and then study its limit distribution.
Global physical properties of random media change qualitatively at a percolation threshold, where isolated clusters merge to form one infinite connected component. The precise knowledge of percolation thresholds is thus of paramount…