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Article Dans Une Revue The Annals of Probability Année : 2022

Scaling and local limits of Baxter permutations and bipolar orientations through coalescent-walk processes

Résumé

Baxter permutations, plane bipolar orientations, and a specific family of walks in the non-negative quadrant, called \emph{tandem walks}, are well-known to be related to each other through several bijections. We introduce a further new family of discrete objects, called \emph{coalescent-walk processes} and we relate it to the three families mentioned above. We prove joint Benjamini--Schramm convergence (both in the annealed and quenched sense) for uniform objects in the four families. Furthermore, we explicitly construct a new random measure of the unit square, called the \emph{Baxter permuton} and we show that it is the scaling limit (in the permuton sense) of uniform Baxter permutations. On top of that, we relate the limiting objects of the four families to each other, both in the local and scaling limit case. The scaling limit result is based on the convergence of the trajectories of the coalescent-walk process to the \emph{coalescing flow} -- in the terminology of Le Jan and Raimond (2004) -- of a perturbed version of the Tanaka stochastic differential equation. Our scaling result entails joint convergence of the tandem walks of a plane bipolar orientation and its dual, extending the main result of Gwynne, Holden, Sun (2016), and answering more precisely Conjecture 4.4 of Kenyon, Miller, Sheffield, Wilson (2019).

Dates et versions

hal-02943343 , version 1 (19-09-2020)

Identifiants

Citer

Jacopo Borga, Mickaël Maazoun. Scaling and local limits of Baxter permutations and bipolar orientations through coalescent-walk processes. The Annals of Probability, 2022, 50 (4), pp.1359-1417. ⟨10.1214/21-AOP1559⟩. ⟨hal-02943343⟩

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