Baxter permutations, plane bipolar orientations, and a specific family of walks in the nonnegative quadrant, called tandem walks, are well-known to be related to each other through several bijections. We introduce a further new family of discrete objects, called 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 on the unit square, called the Baxter permuton and we show that it is the scaling limit (in the permuton sense) of uniform Baxter permutations. In addition, 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 coalescing flow—in the terminology of Le Jan and Raimond (Ann. Probab. 32 (2004) 1247–1315)—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, giving an alternative answer to Conjecture 4.4 of Kenyon, Miller, Sheffield, Wilson (Ann. Probab. 47 (2019) 1240–1269) compared to the one of Gwynne, Holden, Sun ((2016), arXiv:1603.01194).
Thanks to Mathilde Bouvel, Valentin Féray and Grégory Miermont for their dedicated supervision and enlightening discussions. Thanks to Nicolas Bonichon, Nina Holden, Emmanuel Jacob, Jason Miller, Kilian Raschel, Xin Sun, Olivier Raimond and Vitali Wachtel for enriching discussions and pointers. We finally thank the anonymous referees for all their precious and useful comments as well as Emmanuel Kammerer for some remarks. An extended abstract of this preprint is available at .
"Scaling and local limits of Baxter permutations and bipolar orientations through coalescent-walk processes." Ann. Probab. 50 (4) 1359 - 1417, July 2022. https://doi.org/10.1214/21-AOP1559