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Geometry of permutation limits. (English) Zbl 1449.60010
This article concerns permutons (a measure on a square with uniform marginal distributions) and permuton processes (a process $$X$$ with continuous sample paths such that each $$X(t)$$ is uniformly distributed on some fixed interval). In the graph of the symmetric group $$\mathfrak{S}_n$$, a sorting network is a geodesic from the permutation $$(1 2 3 \dots n)$$ to $$(n (n - 1) (n - 2) \dots 1)$$. This article is a hopeful step towards resolving the Archimedean path conjecture of O. Angel et al. [Adv. Math. 215, No. 2, 839–868 (2007; Zbl 1132.60008)], which states that as $$n \to \infty$$, the random sorting network process on $$\mathfrak{S}_n$$ converges in probability to the “Archimedean process” $\mathcal{A}(t) = \cos (\pi t) \mathbb{A}_x + \sin (\pi t) \mathbb{A}_y, t \in [0,1]$ for an appropriate random variable $$(\mathbb{A}_x, \mathbb{A}_y)$$ on the plane, defined in terms of an “Archimedean measure”.
Among permuton processes $$X(t)$$, $$0 \leq t \leq 1$$, on the square $$[-1, 1]$$, such that $$X(1) = - X(0)$$, there is a unique process with minimal Dirichlet energy, and it is Archimedean.
Given a permuton-valued path from the identity permuton (of support $$\{ (x, x) : x \in [-1, 1] \}$$) to the reverse permuton (of support $$\{ (x, -x) : x \in [-1, 1] \}$$), the Dirichlet energy (via the Wasserstein metric) of the path is bounded below by the energy of the Archimedean path, that minimum achieved only by the Archimedean path.