Randomly Oriented Lattices
نویسندگان
چکیده
Geoffrey R. Grimmett Abstract. The square lattice is used to generate an oriented graph in which a rightward or upward arrow is present on each edge with probability a, and a leftward or downward arrow with probability b. Independence between different edges of the square lattice is assumed, but nothing is assumed concerning the dependence between the two possible orientations at any given edge. A property of self-duality is exploited to show that, when a + b = 1, the process is, in a sense to be made precise, either critical or supercritical but not subcritical. This observation enables progress with the percolation problem in which each horizontal edge is oriented rightwards with probability p and otherwise leftwards, and each vertical edge is oriented upwards with probability p and otherwise downwards.
منابع مشابه
N ov 2 00 1 Random walks on randomly oriented lattices 1
Simple random walks on various types of partially horizontally oriented regular lattices are considered. The horizontal orientations of the lattices can be of various types (deterministic or random) and depending on the nature of the orientation the asymptotic behaviour of the random walk is shown to be recurrent or transient. In particular, for randomly horizontally oriented lattices the rando...
متن کاملA Local Limit Theorem for Random Walks in Random Scenery and on Randomly Oriented Lattices
Random walks in random scenery are processes defined by Zn := ∑n k=1 ξX1+...+Xk , where (Xk, k ≥ 1) and (ξy, y ∈ Z) are two independent sequences of i.i.d. random variables. We assume here that their distributions belong to the normal domain of attraction of stable laws with index α ∈ (0, 2] and β ∈ (0, 2] respectively. These processes were first studied by H. Kesten and F. Spitzer, who proved ...
متن کاملh . PR ] 1 5 Ja n 20 02 On the physical relevance of random walks : an example of random walks on a randomly oriented lattice ∗
Random walks on general graphs play an important role in the understanding of the general theory of stochastic processes. Beyond their fundamental interest in probability theory, they arise also as simple models of physical systems. A brief survey of the physical relevance of the notion of random walk on both undirected and directed graphs is given followed by the exposition of some recent resu...
متن کاملCharacterizing trees in property-oriented concept lattices
Property-oriented concept lattices are systems of conceptual clusters called property-oriented concepts, which are partially ordered by the subconcept/superconcept relationships. Property-oriented concept lattices are basic structures used in formal concept analysis. In general, a property-oriented concept lattice may contain overlapping clusters and is not to be a tree construction. Additional...
متن کاملOn Applications of Matroids in Class-oriented Concept Lattices
Class-oriented concept lattices are systems of conceptual clusters, called class-oriented concepts, which are partially ordered by a subconcept-superconcept hierarchy. The hierarchical structure represents a structured information obtained automatically from the input data table. This paper presents the correspondent relations between matroids and class-oriented concept lattices. Under isomorph...
متن کامل