On Limit Constants in Last Passage Percolation in Transitive Tournaments

Abstract

We investigate the \emph{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 $\mathbb{E}[X_n]$, which was stated as an open problem by Yuster [\emph{Disc. Appl. Math.}, 2017]. We further determine scaling constants in the limit laws for $X_n$. Define $\beta_{tr}(p) := \lim_{n\to \infty} \frac{\mathbb{E}[X_n]}{n-1}$. Using singularity analysis, we show \[ \beta_{tr}(p) = \left(\sum_{n\geq 1}(1-p)^{{n\choose 2}}\right)^{-1}. \] In particular, $\beta_{tr}(0.5) = \left(\sum_{n\geq 1} 2^{-{n\choose 2}}\right)^{-1} = 0.60914971106...$. This settles the question of determining the value of $\beta_{tr}(0.5)$, initiated by Yuster. $\beta_{tr}(p)$ is also the limiting value in the strong law of large numbers for $X_n$, given by Foss, Martin, and Schmidt [\emph{Ann. Appl. Probab.}, 2014]. We also derive the scaling constants in the functional central limit theorem for $X_n$ proved by Foss et al.