The Lightning Model

James T. Campbell (), A. Deane () and A. Quas ()
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James T. Campbell: University of Memphis
A. Deane: University of Victoria
A. Quas: University of Victoria

Journal of Theoretical Probability, 2022, vol. 35, issue 4, 2738-2756

Abstract: Abstract We introduce a non-standard model for percolation on the integer lattice $$\mathbb Z^2$$ Z 2 . Randomly assign to each vertex $$a \in \mathbb Z^2$$ a ∈ Z 2 a potential, denoted $$\phi _a$$ ϕ a , chosen independently and uniformly from the interval [0, 1]. For fixed $$\epsilon \in [0,1]$$ ϵ ∈ [ 0 , 1 ] , draw a directed edge from vertex a to a nearest-neighbor vertex b if $$\phi _b p_\text {site}$$ ϵ > p site , the critical probability for standard site percolation in $$\mathbb Z^2$$ Z 2 , the model percolates strongly. We study the number of infinite strongly connected clusters occurring in a typical configuration. We show that for these models of percolation on directed graphs, there are some subtle issues that do not arise for undirected percolation. Although our model does not have the finite energy property, we are able to show that, as in the standard model, the number of infinite strongly connected clusters is almost surely 0, 1 or $$\infty $$ ∞ .

Keywords: Percolation; Integer lattice; Phase transition; Infinite clusters; 60K35; 82B43 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10959-021-01135-z

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