Asymptotic Behaviour of Semi-Infinite Geodesics for Maximal Increasing Subsequences in the Plane
We consider for a given Poissonian cloud ω in ℝ2the maximal up/right path fromxto y(x, y∈ ℝ2), where maximal means that it contains as many points ofω as possible. We prove that with probability 1 there exist for every up/right direction semi-infinite maximal paths (unigeodesics), every unigeodesic has a direction and if there are two unigeodesics with the same direction they must meet and then coalesce. Out of these results we are able to construct a Busemann type function, which behaves at most diffusively.
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