Tournaments and random walks

We study the relationship between tournaments and random walks. This connection was observed by Moser, who conjectured that the number of score sequences Sₙ~ C4ⁿ/n5/2. Winston and Kleitman came close to finding the correct order of Sₙ. Building on this, and subsequent works by Tak\'acs, Kim and Pittel confirmed that Sₙ=Θ(4ⁿ/n5/2). In this work, we verify Moser's conjecture, 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 alternative, 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{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.