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: