Baxter permutations, plane bipolar orientations, and a specific family of walks in the non-negative quadrant are well-known to be related to each other through several bijections. In order to study their scaling and local limits, we introduce a further new family of discrete objects, called coalescent-walk processes and we relate them with the other previously mentioned families introducing some new bijections.
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 fractal random measure of the unit square, called the coalescent Baxter permuton and we show that it is the scaling limit (in the permuton sense) of uniform Baxter permutations.
To prove the latter result, we study the scaling limit of the associated random coalescent-walk processes. We show that they converge in law to a random continuous coalescent-walk process encoded by a perturbed version of the Tanaka stochastic differential equation. This result has connections with the results of Gwynne, Holden, Sun (2016) on scaling limits of plane bipolar triangulations (in the Peanosphere topology). In particular, we prove that uniform bipolar orientations converge to the uniform finite-volume bipolar map. This proves the Conjecture 4.4 of Kenyon, Miller, Sheffield, Wilson (2019) without restricting to triangulations.
We further present some results that relate the limiting objects of the four families to each other, both in the local and scaling limit case.
Joint work with M.Maazoun.
ETH Zürich (Switzerland), Graduate seminar of probability.
CUNY, New York, (USA), online seminar.