Tournaments and random walks

Abstract

We study the relationship between tournaments and random walks. This connection was first observed by Erd\H{o}s and Moser. Winston and Kleitman came close to showing that $S_n=\Theta(4^n/n^{5/2})$. Building on this, and works by Tak\'acs, these asymptotic bounds were confirmed by Kim and Pittel. In this work, we verify Moser's conjecture that $S_n\sim C4^n/n^{5/2}$, using limit theory for integrated random walk bridges. Moreover, we show that $C$ can be described in terms of random walks. Combining this with a recent proof and number-theoretic description of $C$ by the second author, we obtain an analogue of Louchard's formula, for the Laplace transform of the squared Brownian excursion/Airy area measure. Finally, we describe the scaling limit of random score sequences, in terms of the Kolmogorov excursions, studied recently by B\"{a}r, Duraj and Wachtel. Our results can also be interpreted as answering questions related to a class of random polymers, which began with influential work of Sina\u{i}. From this point of view, our methods yield the precise asymptotics of a persistence probability, related to the pinning/wetting models from statistical physics, that was estimated up to constants by Aurzada, Dereich and Lifshits, as conjectured by Caravenna and Deuschel.