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 ⊙. 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 Cρ={Cρβ}β∈[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: Cρβ1⊙ Cρβ2=Cρmin(β1+β2,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 Cρβ equals β/π. For the Archimedean measure Arch the family CArch 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 𝒩 the family C𝒩 consists of Gaussian copulas and contains 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/√2. By the Herschel-Maxwell theorem the Gaussian measure is the only profile whose family contains Unif; nevertheless, 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.
Related
BibTeX
@misc{presniakov2026geodesics,
title = {{Geodesics in the continuous Hecke monoid}},
author = {Sergei Presniakov and Alexander Tiskin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26700},
eprint = {2610.26700},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26700},
}
RIS
TY - UNPB
TI - Geodesics in the continuous Hecke monoid
AU - Sergei Presniakov
AU - Alexander Tiskin
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26700
UR - https://arxiv.org/abs/2610.26700
AB - 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 ⊙. 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∈[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⊙ 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 Arch the family C^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 𝒩 the family C^𝒩 consists of Gaussian copulas and contains 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/√2. By the Herschel-Maxwell theorem the Gaussian measure is the only profile whose family contains Unif; nevertheless, 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.
ER -