Geodesics in the continuous Hecke monoid

Abstract

A permuton is a probability measure on the unit square with uniform marginals, the limit shape of a sequence of permutations. The sticky product of permutons is a continuous analogue of the Demazure product in the $0$-Hecke monoid; on copulas, the distribution functions of permutons, it is given by the tropical $(\min,+)$ product $\odot$. We show that every absolutely continuous rotation-invariant probability measure $ρ$ on the plane, which we call the profile, generates a one-parameter family of copulas $C^ρ=\{C^ρ_β\}_{β\in[0,π]}$. A copula in this family is defined by a pair of projections of a $ρ$-distributed random vector onto directions at angle $π-β$. Along this family, the sticky product adds angles: $$C^ρ_{β_1}\odot C^ρ_{β_2}=C^ρ_{\min(β_1+β_2,π)}.$$ Hence every non-identity element of such a family is a sticky product of two others and has roots of every natural degree. The normalized inversion number of the copula $C^ρ_β$ equals $β/π$. For the Archimedean measure $\mathfrak{Arch}$ the family $C^{\mathfrak{Arch}}$ consists of the limit shapes of the permutations of the uniformly random sorting network, found by Angel et al. and Dauvergne. For the Gaussian measure $\mathcal{N}$ the family $C^{\mathcal{N}}$ consists of Gaussian copulas and contains $\mathit{Unif}$, the uniform measure on the square (the limit shape of the uniformly random permutation); its square root within this family is the Gaussian copula with correlation $-1/\sqrt2$. By the Herschel-Maxwell theorem the Gaussian measure is the only profile whose family contains $\mathit{Unif}$; nevertheless, $\mathit{Unif}$ has another, singular, sticky square root. A discrete analogue of the Gaussian family is the geometric sorting network generated by rotating independent Gaussian points: after rotation by $π/2$ its permutation is exactly uniform.