However, it is well known that for a polyhedron of a special type its volume formula becomes much simpler. Nowadays, it is fully solved for a tetrahedron, the most simple polyhedron in the combinatorial sense. Deriving volume formulas for 3-dimensional non-Euclidean polyhedra of a given combinatorial type is a very difficult problem. This new family of permutons, called \emph$ where $\theta$ is the so-called imaginary geometry angle between a certain pair of SLE curves.Ĭomputation of the volumes of polyhedra is a classical geometry problem known since ancient mathematics and preserving its importance until present time. The method works equally well for a two-parameter generalization of the Baxter permuton recently introduced by the first author, except that the density is not as explicit. Our proofs rely on a recent connection between the Baxter permuton and Liouville quantum gravity (LQG) coupled with the Schramm-Loewner evolution (SLE). We also prove that all pattern densities of the Baxter permuton are strictly positive, distinguishing it from other permutons arising as scaling limits of pattern-avoiding permutations. This answers a question of Dokos and Pak (2014). We find an explict formula for the expectation of the Baxter permuton, i.e.\ the density of its intensity measure. The Baxter permuton is a random probability measure on the unit square which describes the scaling limit of uniform Baxter permutations.
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