Baxter permutations and bipolar orientations

The following images were realized during the preparation of the article “Scaling and local limits of Baxter permutations and bipolar orientations through coalescent-walk processes”.

Here you can see two uniform large Baxter permutations. We showed that they converge (in the permuton sense) to the Baxter permuton, that is a random probability measure of the unit square with uniform marginals.

Baxter permutations are well known to be in bijection with bipolar orientations (a specific family of decorated planar maps) that are, in turn, in bijection with tandem walks (a specific family of two-dimensional walks in the non-negative quadrant). We showed that these three families of discrete objects are also in bijection with some specific coalescent-walk processes. The four images below show four large objects related by these bijections.

The bipolar orientation in the second picture is embedded in the plane respecting the diagram of the corresponding Baxter permutation. We showed that a uniform bipolar orientation converges (in the peanosphere sense) jointly with its dual map  to a \(\sqrt{4/3}\)-Liouville quantum gravity decorated with two orthogonal (in the sense of imaginary geometry) SLE\({}_{12}\) (this result extends previous results of Kenyon, Miller, Sheffield, Wilson and Gwynne, Holden, Sun).

Finally, we show the two trees that form the bipolar orientation: