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

Baxter permutations, plane bipolar orientations, and a specific family of walks in the nonnegative quadrant, called tandem walks, are well-known to be related each other through several bijections. We introduce further new discrete objects, coalescent-walk processes we relate it three families mentioned above. prove joint Benjamini–Schramm convergence (both annealed quenched sense) for uniform objects four families. Furthermore, explicitly construct random measure on unit square, permuton show that is scaling limit (in permutations. In addition, limiting other, both local case. The result based trajectories process coalescing flow—in terminology Le Jan Raimond (Ann. Probab. 32 (2004) 1247–1315)—of perturbed version Tanaka stochastic differential equation. Our entails orientation its dual, giving an alternative answer Conjecture 4.4 Kenyon, Miller, Sheffield, Wilson 47 (2019) 1240–1269) compared one Gwynne, Holden, Sun ((2016), arXiv:1603.01194).