Permutons, which are probability measures on the unit square \([0, 1]^2\) with uniform marginals, are the natural scaling limits for sequences of (random) permutations.
We introduce a \(d\)-dimensional generalization of these measures for all \(d \ge 2\), which we call \(d\)-dimensional permutons, and extend — from the two-dimensional setting — the theory to prove convergence of sequences of (random) \(d\)-dimensional permutations to (random) \(d\)-dimensional permutons.
Building on this new theory, we determine the random high-dimensional permuton limits for two natural families of high-dimensional permutations. First, we determine the \(3\)-dimensional permuton limit for Schnyder wood permutations, which bijectively encode planar triangulations decorated by triples of spanning trees known as Schnyder woods. Second, we identify the \(d\)-dimensional permuton limit for \(d\)-separable permutations, a pattern-avoiding class of \(d\)-dimensional permutations generalizing ordinary separable permutations.
Both high-dimensional permuton limits are random and connected to previously studied universal 2-dimensional permutons, such as the Brownian separable permutons and the skew Brownian permutons, and share interesting connections with objects arising from random geometry, including the continuum random tree, Schramm–Loewner evolutions, and Liouville quantum gravity surfaces.