On Limit Constants in Last Passage Percolation in Transitive Tournaments
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 š¼[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 Ī²_tr(p) := lim_nāāš¼[X_n]/n-1. Using singularity analysis, we show Ī²_tr(p) = (ā_nā„ 1(1-p)^n 2)^-1. In particular, Ī²_tr(0.5) = (ā_nā„ 1 2^-n 2)^-1 = 0.60914971106.... This settles the question of determining the value of Ī²_tr(0.5), initiated by Yuster. Ī²_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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