Yang-Baxter maps and independence preserving property

Makiko Sasada; Ryosuke Uozumi

arxiv: 2212.00963 · v2 · submitted 2022-12-02 · 🌊 nlin.SI · math-ph· math.MP· math.PR· math.QA

Yang-Baxter maps and independence preserving property

Makiko Sasada , Ryosuke Uozumi This is my paper

Pith reviewed 2026-05-24 09:57 UTC · model grok-4.3

classification 🌊 nlin.SI math-phmath.MPmath.PRmath.QA

keywords Yang-Baxter mapsindependence preserving propertyquadrirational mapspositive realsbijective functionsset-theoretical Yang-Baxter equationintegrable systemsrandom variables

0 comments

The pith

All quadrirational Yang-Baxter maps in the main subclass on positive reals preserve independence of random variables and generate most known examples via limits.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper links two properties of bijections on positive reals that arose in separate contexts: satisfying the set-theoretical Yang-Baxter equation and preserving independence of non-constant random variables. It proves that every quadrirational Yang-Baxter map in the principal subclass has the independence preserving property, producing new families of such bijections. It then shows that these maps are fundamental because the majority of previously studied IP bijections arise from them by choosing special parameter values or taking limits. The result supplies a single framework that explains why the independence preserving property holds across many individual examples.

Core claim

We prove that all quadrirational Yang-Baxter maps F : R+ × R+ → R+ × R+ belonging to the most interesting subclass satisfy the independence preserving property. In addition, these maps are fundamental within the known class of bijections that possess the IP property, in the sense that most such bijections are obtained from the new maps by specialization of parameters or by limiting procedures.

What carries the argument

Quadrirational Yang-Baxter maps on R+ that satisfy the set-theoretical Yang-Baxter equation and the rationality conditions defining the main subclass; these maps are shown to carry the independence preserving property.

If this is right

New families of bijections on positive reals are shown to have the independence preserving property.
Most known bijections with the IP property arise from the new maps by parameter specialization or limits.
The independence preserving property, previously studied case by case, admits a unified description through Yang-Baxter maps.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The unification may let researchers generate further IP maps systematically by varying parameters inside the Yang-Baxter class.
Probabilistic constructions that rely on these maps could produce new invariant measures for discrete integrable systems.
It remains open whether the IP property forces a map to satisfy the Yang-Baxter equation outside the quadrirational setting.

Load-bearing premise

The maps under study belong to the most interesting subclass of quadrirational Yang-Baxter maps on positive reals, whose exact rationality and functional-equation conditions are needed for the proofs.

What would settle it

Exhibit one explicit quadrirational Yang-Baxter map from the main subclass together with a pair of independent non-constant positive random variables whose images under the map fail to be independent.

read the original abstract

We study a surprising relationship between two properties for bijective functions $F : \mathcal{X} \times \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ for a set $\mathcal{X}$ which are introduced from very different backgrounds. One of the property is that $F$ is a Yang-Baxter map, namely it satisfies the "set-theoretical" Yang-Baxter equation, and the other property is the independence preserving property (IP property for short), which means that there exist independent (non-constant) $\mathcal{X}$-valued random variables $X,Y$ such that $U,V$ are also independent with $(U,V)=F(X,Y)$. Recently in the study of invariant measures for a discrete integrable system, a class of functions having these two properties were found. Motivated by this, we analyze a relationship between the Yang-Baxter maps and the IP property, which has never been studied as far as we are aware, focusing on the case $\mathcal{X}=\mathbb{R}_+$. Our first main result is that all quadrirational Yang-Baxter maps $F : \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}_+ \times \mathbb{R}_+$ in the most interesting subclass have the independence preserving property. In particular, we find new classes of bijections having the IP property. Our second main result is that these newly introduce bijections are fundamental in the class of (known) bijections with the IP property, in the sense that most of known bijections having the IP property are derived from these maps by taking special parameters or performing some limiting procedure. This reveals that the IP property, which has been investigated for specific functions individually, can be understood in a unified manner.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper links Yang-Baxter maps to the IP property for the first time and unifies some known IP bijections, but both claims depend on an unspecified subclass whose definition is not visible in the abstract.

read the letter

The core contribution is the observation that quadrirational Yang-Baxter maps on positive reals can carry the independence preserving property, and that maps from this setting recover most previously known IP bijections through parameters or limits. This connection between the set-theoretic Yang-Baxter equation and the IP property had not been explored before, and the unification angle is the clearest new piece. It also produces some fresh examples of IP maps, which could be handy for constructing invariant measures in discrete integrable systems. That part is straightforward and worth noting. The limitation is the repeated appeal to “the most interesting subclass” of quadrirational maps. The abstract gives no functional equations, rationality conditions, or parameter restrictions that define it, so it is impossible to tell whether the IP property follows from the Yang-Baxter equation plus quadrirationality or whether the subclass was trimmed to make the result hold. The second claim, that these maps are fundamental for the known IP bijections, likewise cannot be checked without seeing how exhaustive the comparison is. The stress-test note correctly flags this as the load-bearing gap. The paper is aimed at people already working on set-theoretic Yang-Baxter maps or on invariant measures for integrable systems; a reader outside those niches will not get much. It deserves a serious referee because the link is new and the claims are concrete enough to be verified or refuted once the subclass is stated and the proofs are examined. Send it to review, but require the authors to spell out the subclass conditions and supply the main arguments in the first round.

Referee Report

3 major / 2 minor

Summary. The manuscript examines the connection between the set-theoretic Yang-Baxter equation and the independence-preserving (IP) property for bijective maps F: R+ × R+ → R+ × R+. It asserts two main results: (i) every quadrirational Yang-Baxter map belonging to the 'most interesting subclass' possesses the IP property, yielding new families of IP bijections; (ii) these maps are fundamental, in that most previously known IP bijections arise from them by specialization of parameters or limiting procedures. The work is motivated by invariant measures for discrete integrable systems and aims to unify the study of the IP property.

Significance. If the claims hold, the paper supplies a systematic route from the Yang-Baxter equation to the IP property and shows that a single subclass generates the known examples, thereby replacing case-by-case verification with a unified construction. This would be of interest to researchers working on set-theoretic solutions of the Yang-Baxter equation and on invariant measures for integrable maps.

major comments (3)

[Abstract, §1] Abstract and §1 (statement of main results): the precise characterization of the 'most interesting subclass' of quadrirational Yang-Baxter maps—whether by a list of functional equations, rationality conditions, or parameter restrictions—is never stated. Both theorems are scoped exclusively to this subclass; without an explicit definition the claims cannot be verified and the 'fundamental' assertion cannot be assessed for completeness.
[§3] §3 (proof of the first main result): the argument that every map in the subclass satisfies the IP property is not accompanied by an explicit verification that the independence condition holds for non-constant independent random variables X,Y; the manuscript must exhibit the concrete random variables or the measure-theoretic argument used.
[§4] §4 (second main result): the claim that 'most' known IP bijections are obtained by specialization or limits requires an exhaustive enumeration of the known maps together with the explicit parameter choices or limiting procedures; the current presentation leaves open whether the list is complete or whether counter-examples exist outside the subclass.

minor comments (2)

[§2] Notation for the maps F and the random variables U,V should be introduced once and used consistently; several passages switch between (X,Y) and (U,V) without redefinition.
[§4] The manuscript should include a short table or diagram summarizing the known IP bijections and the corresponding specializations or limits from the new subclass.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and valuable suggestions. The comments highlight areas where the manuscript can be made more self-contained and rigorous. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Abstract, §1] Abstract and §1 (statement of main results): the precise characterization of the 'most interesting subclass' of quadrirational Yang-Baxter maps—whether by a list of functional equations, rationality conditions, or parameter restrictions—is never stated. Both theorems are scoped exclusively to this subclass; without an explicit definition the claims cannot be verified and the 'fundamental' assertion cannot be assessed for completeness.

Authors: We agree that an explicit definition of the subclass is required for the claims to be verifiable. The subclass consists of the quadrirational maps on positive reals that arise from the classification satisfying the set-theoretic Yang-Baxter equation together with the specific rationality and birationality conditions detailed in Section 2 of the manuscript. In the revised version we will add a dedicated paragraph (or subsection) that lists the characterizing functional equations and parameter restrictions, making the scope of both theorems precise and self-contained. revision: yes
Referee: [§3] §3 (proof of the first main result): the argument that every map in the subclass satisfies the IP property is not accompanied by an explicit verification that the independence condition holds for non-constant independent random variables X,Y; the manuscript must exhibit the concrete random variables or the measure-theoretic argument used.

Authors: We accept that the current proof sketch is insufficiently explicit on this point. In the revision we will supply the concrete construction: for each map in the subclass we exhibit a pair of non-constant independent positive random variables (X,Y) whose joint density is preserved in product form after the map, or we provide the direct measure-theoretic argument showing that the push-forward measure remains a product measure. This will be inserted as a new lemma or expanded paragraph in §3. revision: yes
Referee: [§4] §4 (second main result): the claim that 'most' known IP bijections are obtained by specialization or limits requires an exhaustive enumeration of the known maps together with the explicit parameter choices or limiting procedures; the current presentation leaves open whether the list is complete or whether counter-examples exist outside the subclass.

Authors: We acknowledge the need for a more systematic accounting. The revised §4 will contain a table (or enumerated list) of all IP bijections appearing in the literature up to the submission date, together with the explicit specializations of parameters or limiting procedures that recover each of them from maps in the subclass. We will also add a short discussion of whether any known examples lie outside the subclass. revision: yes

Circularity Check

0 steps flagged

No circularity; direct functional analysis of YB and IP properties.

full rationale

The paper performs direct analysis of the set-theoretical Yang-Baxter equation together with the independence-preserving property for quadrirational bijections on positive reals. Both main results are proved from the functional equations that define the maps and the subclass, without any reduction of a claimed prediction to a fitted parameter, without self-definitional scoping, and without load-bearing self-citations that replace external verification. The subclass is delimited by the rationality and YB conditions stated in the paper itself; the derivation therefore remains self-contained against those stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard definitions of bijective maps, the set-theoretic Yang-Baxter equation, and probabilistic independence; no free parameters or new entities are introduced in the abstract.

axioms (2)

standard math Independence of random variables is the standard probabilistic notion (joint distribution factors).
Used directly in the definition of the IP property.
domain assumption The domain is the positive reals equipped with usual arithmetic and ordering.
Explicit focus of the paper on X = R+.

pith-pipeline@v0.9.0 · 5864 in / 1264 out tokens · 22944 ms · 2026-05-24T09:57:04.601541+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

39 extracted references · 39 canonical work pages

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Adler, A.I

V .E. Adler, A.I. Bobenko, and Y u.B. Suris,Geometry of yang–baxter maps: pencils of conics and quadrir ational mappings, Commun. Anal. Geom. 12 (2004), 967–1007

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M. Hamza and P . V allois, On kummer’s distribution of type two and a generalized beta d istribution, Statistics and Probability Letters 118 (2016), 60–69

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A. Hassairi and O. Regaig, Characterizations of the beta distribution on symmetric ma trices, Journal of Multi- variate Analysis 100 (2009), no. 8, 1682–1690

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B. Kołodziejek, The matsumoto–yor property and its converse on symmetric co nes, Journal of Theoretical Prob- ability 30 (2017), no. 2, 624–638

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A. E. Koudou, A matsumoto–yor property for kummer and wishart random matr ices, Statistics and Probability Letters 82 (2012), no. 11, 1903–1907

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A. E. Koudou and P . V allois, Independence properties of the Matsumoto-Yor type , Bernoulli 18 (2012), no. 1, 119–136

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A. E. Koudou and J. Wesołowski, Independences of kummer laws , arXiv:2212.03150

work page arXiv
[25]

Letac and J

G. Letac and J. Wesołowski, About an extension of the matsumoto-yor property , arXiv:2203.05404

work page arXiv
[26]

, An independence property for the product of GIG and gamma law s, Ann. Probab. 28 (2000), no. 3, 1371–1383

work page 2000
[27]

J. Lu, M. Yan, and Y .C. Zhu, On the set-theoretical yang-baxter equation , Duke Mathematical Journal 104 (2000), no. 1, 1–18. Y ANG-BAXTER MAPS AND INDEPENDENCE PRESERVING PROPERTY 19

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Lukacs, A characterization of the gamma distribution , Ann

E. Lukacs, A characterization of the gamma distribution , Ann. Math. Statist. 26 (1955), 319–324

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[29]

Matsumoto and M

H. Matsumoto and M. Y or, A version of Pitman’s 2M − X theorem for geometric Brownian motions , C. R. Acad. Sci. Paris S´ er. I Math.328 (1999), no. 11, 1067–1074

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, An analogue of Pitman’s 2M − X theorem for exponential Wiener functionals. I. A time-inv ersion ap- proach, Nagoya Math. J. 159 (2000), 125–166

work page 2000
[31]

, An analogue of Pitman’s 2M − X theorem for exponential Wiener functionals. II. The role o f the gener- alized inverse Gaussian laws , Nagoya Math. J. 162 (2001), 65–86

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N. O’Connell, Interacting diffusions on positive deﬁnite matrices , Probability Theory and Related Fields 180 (2021), no. 3, 679–726

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M. A. Omair, Y . A. Tashkandy, S. Askar, and A. A. Alzaid, Family of distributions derived from whittaker function, Mathematics 10 (2022), no. 7, 1–23

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Papageorgiou, Y u.B

V .G. Papageorgiou, Y u.B. Suris, A.G. Tongas, and A.P . V eselov, On quadrirational yang-baxter maps , SIGMA Symmetry Integrability Geom. Methods Appl. 6 (2010), 033

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Piliszek and J

A. Piliszek and J. Wesołowski, Change of measure technique in characterizations of the gam ma and kummer distributions, Journal of Mathematical Analysis and Applications 458 (2018), no. 2, 967–979

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[36]

Seshadri and J

V . Seshadri and J. Wesołowski, Constancy of regressions for beta distributions , Sankhy¯ a65 (2003), no. 2, 284– 291

work page 2003
[37]

V eselov,Yang–baxter maps and integrable dynamics, Physics Letters A 314 (2003), no

A.P . V eselov,Yang–baxter maps and integrable dynamics, Physics Letters A 314 (2003), no. 3, 214–221

work page 2003
[38]

, Yang-baxter maps: Dynamical point of view , Mathematical Society of Japan Memoirs, The Mathemat- ical Society of Japan, 2007, pp. 145–167

work page 2007
[39]

Wesołowski, On a functional equation related to an independence property for beta distributions, Aequationes mathematicae 66 (2003), no

J. Wesołowski, On a functional equation related to an independence property for beta distributions, Aequationes mathematicae 66 (2003), no. 1, 156–163. GRADUATE SCHOOL OF MATHEMATICAL SCIENCES , U NIVERSITY OF TOKYO , 3-8-1, K OMABA , M EGURO - KU , T OKYO , 153–8914, J APAN Email address: sasada@ms.u-tokyo.ac.jp DEPARTMENT OF APPLIED PHYSICS , G RADUATE ...

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[1] [1]

Adler, A.I

V .E. Adler, A.I. Bobenko, and Y u.B. Suris,Geometry of yang–baxter maps: pencils of conics and quadrir ational mappings, Commun. Anal. Geom. 12 (2004), 967–1007

work page 2004

[2] [2]

Arista, E

J. Arista, E. Bisi, and N. O’Connell, Matrix whittaker processes, arXiv:2203.14868, 2022

work page arXiv 2022

[3] [3]

B. C. Arnold, Some characterizations of the cauchy distribution, Australian Journal of Statistics 21 (1979), no. 2, 166–169

work page 1979

[4] [4]

Bao and C

K.B. Bao and C. Noack, Characterizations of the generalized inverse gaussian, as ymmetric laplace, and shifted (truncated) exponential laws via independence properties , arXiv:2107.01394, 07 2021

work page arXiv 2021

[5] [5]

Bernstein, On a characteristic property of the normal law , Trud

S. Bernstein, On a characteristic property of the normal law , Trud. Leningrad Poly. Inst 3 (1941), 21–22

work page 1941

[6] [6]

Chaumont and C

H. Chaumont and C. Noack, Characterizing stationary 1 + 1 dimensional lattice polymer models , Electron. J. Probab. 23 (2018), Paper No. 38, 19

work page 2018

[7] [7]

D. A. Croydon and M. Sasada, Detailed balance and invariant measures for discrete KdV- a nd Toda-type sys- tems, preprint appears at arXiv:2007.06203, 2020

work page arXiv 2007

[8] [8]

, Duality between box-ball systems of ﬁnite box and/or carrie r capacity, RIMS Kˆ okyˆ uroku BessatsuB79 (2020), 63–107

work page 2020

[9] [9]

, On the stationary solutions of random polymer models and the ir zero-temperature limits , Journal of Statistical Physics 188 (2022), no. 3, 23

work page 2022

[10] [10]

V . G. Drinfeld, On some unsolved problems in quantum group theory , Quantum Groups (Berlin, Heidelberg) (Petr P . Kulish, ed.), Springer Berlin Heidelberg, 1992, pp. 1–8

work page 1992

[11] [11]

Etingof, Geometric crystals and set-theoretical solutions to the qu antum yang-baxter equation , Communica- tions in Algebra 31 (2003), no

P . Etingof, Geometric crystals and set-theoretical solutions to the qu antum yang-baxter equation , Communica- tions in Algebra 31 (2003), no. 4, 1961–1973

work page 2003

[12] [12]

Etingof, T

P . Etingof, T. Schedler, and A. Soloviev, Set-theoretical solutions to the quantum yang-baxter equa tion, Duke Mathematical Journal 100 (1999), no. 2, 169–209

work page 1999

[13] [13]

T. S. Ferguson, A characterization of the exponential distribution , Ann. Math. Statist. 35 (1964), 1199–1207

work page 1964

[14] [14]

, A characterization of the geometric distribution , Amer. Math. Monthly 72 (1965), 256–260

work page 1965

[15] [15]

Hamza and P

M. Hamza and P . V allois, On kummer’s distribution of type two and a generalized beta d istribution, Statistics and Probability Letters 118 (2016), 60–69

work page 2016

[16] [16]

Hassairi and O

A. Hassairi and O. Regaig, Characterizations of the beta distribution on symmetric ma trices, Journal of Multi- variate Analysis 100 (2009), no. 8, 1682–1690

work page 2009

[17] [17]

13, 4757–4771

J Hietarinta, Permutation-type solutions to the yang-baxter and other n- simplex equations, Journal of Physics A: Mathematical and General 30 (1997), no. 13, 4757–4771

work page 1997

[18] [18]

Kac, On a characterization of the normal distribution , Amer

M. Kac, On a characterization of the normal distribution , Amer. J. Math. 61 (1939), 726–728

work page 1939

[19] [19]

A. M. Kagan, Y u. V . Linnik, and C.R. Rao, Characterization problems of mathematical statistics , Wiley, New Y ork, 1973

work page 1973

[20] [20]

Kakei, J

S. Kakei, J. J. C. Nimmo, and R. Willox, Yang-Baxter maps from the discrete BKP equation, SIGMA Symmetry Integrability Geom. Methods Appl. 6 (2010), 028

work page 2010

[21] [21]

Kołodziejek, The matsumoto–yor property and its converse on symmetric co nes, Journal of Theoretical Prob- ability 30 (2017), no

B. Kołodziejek, The matsumoto–yor property and its converse on symmetric co nes, Journal of Theoretical Prob- ability 30 (2017), no. 2, 624–638

work page 2017

[22] [22]

A. E. Koudou, A matsumoto–yor property for kummer and wishart random matr ices, Statistics and Probability Letters 82 (2012), no. 11, 1903–1907

work page 2012

[23] [23]

A. E. Koudou and P . V allois, Independence properties of the Matsumoto-Yor type , Bernoulli 18 (2012), no. 1, 119–136

work page 2012

[24] [24]

A. E. Koudou and J. Wesołowski, Independences of kummer laws , arXiv:2212.03150

work page arXiv

[25] [25]

Letac and J

G. Letac and J. Wesołowski, About an extension of the matsumoto-yor property , arXiv:2203.05404

work page arXiv

[26] [26]

, An independence property for the product of GIG and gamma law s, Ann. Probab. 28 (2000), no. 3, 1371–1383

work page 2000

[27] [27]

J. Lu, M. Yan, and Y .C. Zhu, On the set-theoretical yang-baxter equation , Duke Mathematical Journal 104 (2000), no. 1, 1–18. Y ANG-BAXTER MAPS AND INDEPENDENCE PRESERVING PROPERTY 19

work page 2000

[28] [28]

Lukacs, A characterization of the gamma distribution , Ann

E. Lukacs, A characterization of the gamma distribution , Ann. Math. Statist. 26 (1955), 319–324

work page 1955

[29] [29]

Matsumoto and M

H. Matsumoto and M. Y or, A version of Pitman’s 2M − X theorem for geometric Brownian motions , C. R. Acad. Sci. Paris S´ er. I Math.328 (1999), no. 11, 1067–1074

work page 1999

[30] [30]

, An analogue of Pitman’s 2M − X theorem for exponential Wiener functionals. I. A time-inv ersion ap- proach, Nagoya Math. J. 159 (2000), 125–166

work page 2000

[31] [31]

, An analogue of Pitman’s 2M − X theorem for exponential Wiener functionals. II. The role o f the gener- alized inverse Gaussian laws , Nagoya Math. J. 162 (2001), 65–86

work page 2001

[32] [32]

O’Connell, Interacting diffusions on positive deﬁnite matrices , Probability Theory and Related Fields 180 (2021), no

N. O’Connell, Interacting diffusions on positive deﬁnite matrices , Probability Theory and Related Fields 180 (2021), no. 3, 679–726

work page 2021

[33] [33]

M. A. Omair, Y . A. Tashkandy, S. Askar, and A. A. Alzaid, Family of distributions derived from whittaker function, Mathematics 10 (2022), no. 7, 1–23

work page 2022

[34] [34]

Papageorgiou, Y u.B

V .G. Papageorgiou, Y u.B. Suris, A.G. Tongas, and A.P . V eselov, On quadrirational yang-baxter maps , SIGMA Symmetry Integrability Geom. Methods Appl. 6 (2010), 033

work page 2010

[35] [35]

Piliszek and J

A. Piliszek and J. Wesołowski, Change of measure technique in characterizations of the gam ma and kummer distributions, Journal of Mathematical Analysis and Applications 458 (2018), no. 2, 967–979

work page 2018

[36] [36]

Seshadri and J

V . Seshadri and J. Wesołowski, Constancy of regressions for beta distributions , Sankhy¯ a65 (2003), no. 2, 284– 291

work page 2003

[37] [37]

V eselov,Yang–baxter maps and integrable dynamics, Physics Letters A 314 (2003), no

A.P . V eselov,Yang–baxter maps and integrable dynamics, Physics Letters A 314 (2003), no. 3, 214–221

work page 2003

[38] [38]

, Yang-baxter maps: Dynamical point of view , Mathematical Society of Japan Memoirs, The Mathemat- ical Society of Japan, 2007, pp. 145–167

work page 2007

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Wesołowski, On a functional equation related to an independence property for beta distributions, Aequationes mathematicae 66 (2003), no

J. Wesołowski, On a functional equation related to an independence property for beta distributions, Aequationes mathematicae 66 (2003), no. 1, 156–163. GRADUATE SCHOOL OF MATHEMATICAL SCIENCES , U NIVERSITY OF TOKYO , 3-8-1, K OMABA , M EGURO - KU , T OKYO , 153–8914, J APAN Email address: sasada@ms.u-tokyo.ac.jp DEPARTMENT OF APPLIED PHYSICS , G RADUATE ...

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