Inhomogeneous Percolation on Ladder Graphs

Réka Szabó () and Daniel Valesin ()
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Réka Szabó: University of Groningen
Daniel Valesin: University of Groningen

Journal of Theoretical Probability, 2020, vol. 33, issue 2, 992-1010

Abstract: Abstract We define an inhomogeneous percolation model on “ladder graphs” obtained as direct products of an arbitrary graph $$G = (V,E)$$G=(V,E) and the set of integers $${\mathbb {Z}}$$Z. (Vertices are thought of as having a “vertical” component indexed by an integer.) We make two natural choices for the set of edges, producing an unoriented graph $${\mathbb {G}}$$G and an oriented graph $$\vec {{\mathbb {G}}}$$G→. These graphs are endowed with percolation configurations in which independently, edges inside a fixed infinite “column” are open with probability q and all other edges are open with probability p. For all fixed q one can define the critical percolation threshold $$p_\mathrm{c}(q)$$pc(q). We show that this function is continuous in (0, 1).

Keywords: Inhomogeneous percolation; Oriented percolation; Ladder graphs; Critical parameter; 60K35; 82B43 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s10959-019-00896-y

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