On Limit Constants in Last Passage Percolation in Transitive Tournaments
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We investigate the last passage percolation problem on transitive tournaments, in the case when the edge weights are independent Bernoulli random variables. Given a transitive tournament on n nodes with random weights on its edges, the last passage percolation problem seeks to find the weight X_n of the heaviest path, where the weight of a path is the sum of the weights on its edges. We give a recurrence relation and use it to obtain a (bivariate) generating function for the probability generating function of X_n. This also gives exact combinatorial expressions for 𝔼[X_n], which was stated as an open problem by Yuster [Disc. Appl. Math., 2017]. We further determine scaling constants in the limit laws for X_n. Define β_tr(p) := lim_n→∞𝔼[X_n]/n-1. Using singularity analysis, we show β_tr(p) = (∑_n≥ 1(1-p)^n 2)^-1. In particular, β_tr(0.5) = (∑_n≥ 1 2^-n 2)^-1 = 0.60914971106.... This settles the question of determining the value of β_tr(0.5), initiated by Yuster. β_tr(p) is also the limiting value in the strong law of large numbers for X_n, given by Foss, Martin, and Schmidt [Ann. Appl. Probab., 2014]. We also derive the scaling constants in the functional central limit theorem for X_n proved by Foss et al.
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