Sina\u{i} excursions: An analogue of Sparre Andersen's formula for the area process of a random walk

Sina\u{i} initiated the study of random walks with persistently positive area processes, motivated by shock waves in solutions to the inviscid Burgers'equation. We find the precise asymptotic probability that the area process of a random walk bridge is an excursion. A key ingredient is an analogue of Sparre Andersen's classical formula. The asymptotics are related to von Sterneck's subset counting formulas from additive number theory. Our results sharpen bounds by Aurzada, Dereich and Lifshits and respond to a question of Caravenna and Deuschel, which arose in their study of the wetting model. In this context, Sina\u{\i} excursions are a class of random polymer chains exhibiting entropic repulsion.