The critical probability for Voronoi percolation in the hyperbolic plane tends to 1/2

We consider percolation on the Voronoi tessellation generated by a homogeneous Poisson point process on the hyperbolic plane. We show that the critical probability for the existence of an infinite cluster tends to 1/2 as the intensity of the Poisson process tends to infinity. This confirms a conjecture of Benjamini and Schramm [5].     

Percolation on dual lattices with k‐fold symmetry

Zhang found a simple, elegant argument deducing the nonexistence of an infinite open cluster in certain lattice percolation models (for example, p = 1/2 bond percolation on the square lattice) from general results on the uniqueness of an infinite open cluster when it exists; this argument requires some symmetry. Here we show that a simple modification of Zhang's argument requires only two‐fold (or three‐fold) symmetry, proving that the critical probabilities for percolation on dual planar lattices with such symmetry sum to 1. Like Zhang's argument, our extension applies in many contexts; in particular, it enables us to answer a question of Grimmett concerning the anisotropic random cluster model on the triangular lattice. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Enlargement of subgraphs of infinite graphs by Bernoulli percolation

We consider changes in properties of a subgraph of an infinite graph resulting from the addition of open edges of Bernoulli percolation on the infinite graph to the subgraph. We give the triplet of an infinite graph, one of its subgraphs, and a property of the subgraphs. Then, in a manner similar to the way Hammersley’s critical probability is defined, we can define two values associated with the triplet. We regard the two values as certain critical probabilities, and compare them with Hammersley’s critical probability. In this paper, we focus on the following cases of a graph property: being a transient subgraph, having finitely many cut points or no cut points, being a recurrent subset, or being connected. Our results depend heavily on the choice of the triplet.Most results of this paper are announced in Okamura (2016) [24] without proofs. This paper gives full details of them.     

Monotonicity and phase diagram for multirange percolation on oriented trees

We consider Bernoulli bond percolation on oriented regular trees, where besides the usual short bonds, all bonds of a certain length are added. Independently, short bonds are open with probability p and long bonds are open with probability q. We study properties of the critical curve which delimits the set of pairs (p,q) for which there are almost surely no infinite paths. We also show that this curve decreases with respect to the length of the long bonds.     

Constructive equivalence relations on computable probability measures

A central object of study in the field of algorithmic randomness are notions of randomness for sequences, i.e., infinite sequences of zeros and ones. These notions are usually defined with respect to the uniform measure on the set of all sequences, but extend canonically to other computable probability measures. This way each notion of randomness induces an equivalence relation on the computable probability measures where two measures are equivalent if they have the same set of random sequences.In what follows, we study the equivalence relations induced by Martin-Löf randomness, computable randomness, Schnorr randomness and Kurtz randomness, together with the relations of equivalence and consistency from probability theory. We show that all these relations coincide when restricted to the class of computable strongly positive generalized Bernoulli measures. For the case of arbitrary computable measures, we obtain a complete and somewhat surprising picture of the implications between these relations that hold in general.     

Clairvoyant scheduling of random walks

Two infinite walks on the same finite graph are called compatible if it is possible to introduce delays into them in such a way that they never collide. Years ago, Peter Winkler asked the question: for which graphs are two independent random walks compatible with positive probability. Up to now, no such graphs were found. We show in this paper that large complete graphs have this property. The question is equivalent to a certain dependent percolation with a power‐law behavior: the probability that the origin is blocked at distance n but not closer decreases only polynomially fast and not, as usual, exponentially. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

Z~2上相依渗流开簇的极限定理(英文)

中心极限定理, 大偏差定理和大数定律等极限定理在概率论中起着很重要的角色. 本文我们研究Z2上一类相依渗流模型. 对此模型, 我们不仅证明了其无穷开簇的存在唯一性, 而且得到了关于格点盒子类极大开簇的中心极限定理.     

Graph diameter in long‐range percolation

We study the asymptotic growth of the diameter of a graph obtained by adding sparse “long” edges to a square box in ${\mathbb Z}^dWe study the asymptotic growth of the diameter of a graph obtained by adding sparse “long” edges to a square box in ${\mathbb Z}^d$. 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| → ∞. For s ∈ (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 := log₂(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 the intrinsic metric on the (infinite) graph will roughly coincide with a ball of radius exp{r^1/Δ+o(1)} in the Euclidean metric. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 39, 210‐227, 2011     

The cycle space of a 3-connected locally finite graph is generated by its finite and infinite peripheral circuits

We extend Tutte's result that in a finite 3-connected graph the cycle space is generated by the peripheral circuits to locally finite graphs. Such a generalization becomes possible by the admission of infinite circuits in the graph compactified by its ends.     

Heat Kernel Estimates for Strongly Recurrent Random Walk on Random Media

We establish general estimates for simple random walk on an arbitrary infinite random graph, assuming suitable bounds on volume and effective resistance for the graph. These are generalizations of the results in Barlow et al. (Commun. Math. Phys. 278:385–431, 2008, Sects. 1, 2) and in particular imply the spectral dimension of the random graph. We will also give an application of the results to random walk on a long-range percolation cluster. J. Misumi research partially supported by the 21 century COE program at Graduate School of Mathematical Sciences, the University of Tokyo.     

On the Monotonicity of the Speed of Random Walks on a Percolation Cluster of Trees

The authors consider the simple random walk on the infinite cluster of the Bernoulli bond percolation of trees, and investigate the relation between the speed of the simple random walk and the retaining probability p by studying three classes of trees. A sufficient condition is established for Galton-Watson trees.     

Families of Dot-Product Snarks on Orientable Surfaces of Low Genus

We introduce a generalized dot product and provide some embedding conditions under which the genus of a graph does not rise under a dot product with the Petersen graph. Using these conditions, we disprove a conjecture of Tinsley and Watkins on the genus of dot products of the Petersen graph and show that both Grünbaum’s Conjecture and the Berge-Fulkerson Conjecture hold for certain infinite families of snarks. Additionally, we determine the orientable genus of four known snarks and two known snark families, construct a new example of an infinite family of snarks on the torus, and construct ten new examples of infinite families of snarks on the 2-holed torus; these last constructions allow us to show that there are genus-2 snarks of every even order n ≥ 18.     

Infinite paths in planar graphs IV,dividing cycles

Nash‐Williams conjectured that a 4‐connected infinite planar graph contains a spanning 2‐way infinite path if, and only if, the deletion of any finite set of vertices results in at most two infinite components. In this article, we prove this conjecture for graphs with no dividing cycles and for graphs with infinitely many vertex disjoint dividing cycles. A cycle in an infinite plane graph is called dividing if both regions of the plane bounded by this cycle contain infinitely many vertices of the graph. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 173–195, 2006     

Rectangular and visibility representations of infinite planar graphs

We provide a new method for extending results on finite planar graphs to the infinite case. Thus a result of Ungar on finite graphs has the following extension: Every infinite, planar, cubic, cyclically 4‐edge‐connected graph has a representation in the plane such that every edge is a horizontal or vertical straight line segment, and such that no two edges cross. A result of Tamassia and Tollis extends as follows: Every countably infinite planar graph is a subgraph of a visibility graph. Furthermore, every locally finite, 2‐connected, planar graph is a visibility graph. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 257–265, 2006     

Hitting time results for Maker‐Breaker games

We study Maker‐Breaker games played on the edge set of a random graph. Specifically, we analyze the moment a typical random graph process first becomes a Maker's win in a game in which Maker's goal is to build a graph which admits some monotone increasing property \begin{align*}\mathcal{P}\end{align*}. We focus on three natural target properties for Maker's graph, namely being k ‐vertex‐connected, admitting a perfect matching, and being Hamiltonian. We prove the following optimal hitting time results: with high probability Maker wins the k ‐vertex connectivity game exactly at the time the random graph process first reaches minimum degree 2k; with high probability Maker wins the perfect matching game exactly at the time the random graph process first reaches minimum degree 2; with high probability Maker wins the Hamiltonicity game exactly at the time the random graph process first reaches minimum degree 4. The latter two statements settle conjectures of Stojakovi? and Szabó. We also prove generalizations of the latter two results; these generalizations partially strengthen some known results in the theory of random graphs. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

Randomly coloring sparse random graphs with fewer colors than the maximum degree

We analyze Markov chains for generating a random k‐coloring of a random graph G_n,d/n. When the average degree d is constant, a random graph has maximum degree Θ(log n/log log n), with high probability. We show that, with high probability, an efficient procedure can generate an almost uniformly random k‐coloring when k = Θ(log log n/log log log n), i.e., with many fewer colors than the maximum degree. Previous results hold for a more general class of graphs, but always require more colors than the maximum degree. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Entanglement in Percolation

We study finite and infinite entangled graphs in the bond percolationprocess in three dimensions with density p. After a discussionof the relevant definitions, the entanglement critical probabilitiesare defined. The size of the maximal entangled graph at theorigin is studied for small p, and it is shown that this graphhas radius whose tail decays at least as fast as exp(–n/logn); indeed, the logarithm may be replaced by any iterate oflogarithm for an appropriate positive constant . We explorethe question of almost sure uniqueness of the infinite maximalopen entangled graph when p is large, and we establish two relevanttheorems. We make several conjectures concerning the propertiesof entangled graphs in percolation. http://www.statslab.cam.ac.uk/\simgrg/1991 Mathematics Subject Classification: primary 60K35; secondary05C10, 57M25, 82B41, 82B43, 82D60.     

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

In (Aldous, Math. Proc. Cambridge Philos. Soc. 128 (2000), 465–477), 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 →∞. For instance, if one looks at the cluster of a given vertex, does, as N →∞, the probability that it eventually freezes go to 1? Does this probability go to 0? More generally, what can be said about the size of a final cluster? We give a partial answer to some of such questions. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

A Ramsey-type theorem for traceable graphs

A graph G is traceable if there is a path passing through all the vertices of G. It is proved that every infinite traceable graph either contains arbitrarily large finite chordless paths, or contains a subgraph isomorphic to graph A, illustrated in the text. A corollary is that every finitely generated infinite lattice of length 3 contains arbitrarily large finite fences. It is also proved that every infinite traceable graph containing no chordless four-point path contains a subgraph isomorphic to K_ω,ω. The versions of these results for finite graphs are discussed.