Geodesics in the continuous Hecke monoid

Sergei Presniakov, Alexander Tiskin

Oct 19, 2026arXiv:2610.26700v1

math.PRmath.CO

Scorecard· 16/16

6.5/10 impact

Elegant exact angle-addition law unifying sorting-network limit shapes, copulas and the sticky product, with sharp rigidity/non-uniqueness results; confined to a niche subfield and technically elementary.

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 $C_{β}^{ρ} C^ρ_β$ 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.

AI Impact Assessments

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Core Contribution

The paper studies the "sticky product" of permutons, a continuous analogue of the Demazure product in the 0-Hecke monoid. On copulas this product is the tropical (min,+) product. The main result (Theorem 4.1) is a clean angle-addition law. Take any absolutely continuous rotation-invariant planar probability measure ρ (the "profile"). Let C^ρ_β be the copula of the pair of projections (X_0, −X_β). Then C^ρ_{β1} ⊙ C^ρ_{β2} = C^ρ_{min(β1+β2, π)}.

Consequences:

Each such family is a one-parameter "geodesic" from the identity W to the absorbing element M.

The parameter β/π equals the normalized inversion number.

Every non-identity element has roots of every degree within the family.

The Archimedean family, which gives the limit shapes of uniformly random sorting networks (Angel–Holroyd–Romik–Virág, Dauvergne), turns out to be a sticky-product geodesic.

The Gaussian family consists of Gaussian copulas and passes through Unif at β=π/2. This gives explicit sticky roots of the uniform permuton, with correlation −cos(π/2n).

Herschel–Maxwell rigidity shows that only the Gaussian profile passes through Unif.

Unif nevertheless has a singular sticky square root Q, so roots are not unique globally.

A geometric sorting network built from rotating i.i.d. Gaussian points is exactly uniform at angle π/2.

Methodological Rigor

The proofs are elementary and verifiable.

The angle-addition proof reduces the min over t to an infimum over s of a function of three half-planes determined by one random vector G. The key identity B = λA − μC, together with sector-sidedness arguments, makes the interior and absorption regimes transparent. This is a neat argument.

The rigidity theorem is a short reduction to Herschel–Maxwell, and a self-contained characteristic-function derivation is included.

The singular root Q is verified by explicit copula computation: concavity of C_{P*}(t,w), then a case analysis of a subgradient condition. The case analysis is partly sketched, with density and continuity used to cover boundary cases, but it looks correct.

The paper is honest about what is known (Theorem 8.1, the subnetwork theorem) and what is new.

There are no numerical experiments beyond illustrative figures. None are needed, since the results are exact.

Potential Impact

This is a theory paper in probability and algebraic combinatorics. Its value is conceptual: it reveals hidden algebraic structure, namely a one-parameter semigroup (a "logarithm") inside the continuous Hecke monoid. It also links random sorting networks, copulas, tropical algebra and Gaussian geometry. Likely effects:

It will be cited by work on permuton/Demazure products (Defant, Tikhonov and others), on random sorting networks, and on tropical/min-plus structures in KPZ-type models.

It poses well-motivated open questions: whether Mallows permutons form a geodesic, whether an intrinsic Gaussian sorting network exists, and whether there is a q-deformation.

Applied relevance is indirect, through sequence alignment and the semi-local LCS algorithms by Tiskin, but the paper does not pursue this.

Impact will be confined mostly to a niche community, but it is a clean result in a young subfield.

Timeliness & Relevance

Permuton limits of Demazure products are a very recent topic (Defant 2025, Pflueger, the authors' own earlier work). Dauvergne's Archimedean limit is a major recent result. Connecting them is timely, and the paper offers a neat structural explanation: the sorting-network limit shapes compose additively under the sticky product. The comparison with the oriented swap semigroup and with directed-landscape metric composition situates the work in the current KPZ and integrable-probability context.

Strengths

The central theorem is short, exact and elegant, with a transparent geometric proof.

It gives a structural reinterpretation of a deep result (the Archimedean limit) as a semigroup property.

Explicit roots of the uniform permuton (correlation −1/√2) are a memorable concrete outcome.

The rigidity and non-uniqueness results delineate exactly what the projection construction can and cannot do.

It observes that Gaussian geometric networks are exactly uniform at π/2.

The open-questions section is rich.

Limitations

The approach is restricted to projection families. Rigidity shows these cannot capture other natural paths, such as Mallows or the oriented swap process (λ_α, which has singular parts).

The results do not give new information about random sorting networks themselves. Corollary 8.3 is essentially a corollary of the angle-addition theorem plus known limit shapes.

The geometric Gaussian network is not an intrinsic random sorting network.

The link to the arctic ellipse is only an observation.

Ties to applications are absent.

Mathematically it is not deep. The proofs are elementary, and the contribution is more insight and framing than technical difficulty.

Other Observations

The paper is well organized with clear statements, and dependencies are noted explicitly. The notation is somewhat heavy: the W/M conventions, the reflection between the matrix and graph identifications, and the "Archimedean" angle conventions all require care. It is fully self-contained, so reproducibility of the proofs is high. The paper is not a general breakthrough, but it is a polished, insightful contribution likely to be cited as the reference for "geodesics" in the permuton Hecke monoid.

Rating:6.5/ 10

Significance 6Rigor 8Novelty 7Clarity 7.5

Generated Oct 20, 2026

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Scorecard· 16/16

6.5/10 impact