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Consider the graph obtained by superposition of an independent pair of uniform infinite non-crossing perfect matchings of the set of integers. We prove that this graph contains at most one infinite path. Several motivations are discussed.
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Nicolas Curien, Gady Kozma, Vladas Sidoravicius, Laurent Tournier, Uniqueness of the infinite noodle. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 6 (2019), no. 2, pp. 221–238