Abstract
L. Dubins conjectured in 1984 that the graph on vertices {1, 2, 3, ...} where an edge is drawn between verticesi andj with probabilityp ij=λ/max(i, j) independently for each pairi andj is a.s. connected forλ=1. S. Kalikow and B. Weiss proved that the graph is a.s. connected for anyλ>1. We prove Dubin’s conjecture and show that the graph is a.s. connected for anyλ>1/4. We give a proof based on a recent combinatorial result that forλ≦1/4 the graph is a.s. disconnected. This was already proved forλ<1/4 by Kalikow and Weiss. Thusλ=1/4 is the critical value for connectedness, which is surprising since it was believed that the critical value is atλ=1.
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Shepp, L.A. Connectedness of certain random graphs. Israel J. Math. 67, 23–33 (1989). https://doi.org/10.1007/BF02764896
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DOI: https://doi.org/10.1007/BF02764896