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The full persistence probability of a one-dimensional Ising spin is a Painlevé VI function, and the known persistence exponent is the asymptotic mean curvature of a Bonnet surface.

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2026-08-02 17:05 UTC pith:XQXYX4Y5

load-bearing objection Full persistence distribution claimed as a PVI with Manin coefficients; machinery solid, but the key probabilistic identity is borrowed from an unpublished note and the paper does not prove it. the 3 major comments →

arxiv 2603.28632 v1 pith:XQXYX4Y5 submitted 2026-03-30 math-ph cond-mat.stat-mechmath.CAmath.DGmath.MPmath.PR

Geometry of the Ising persistence problem and the universal Bonnet-Manin Painlev\'e VI distribution

Ivan Dornic , Robert Conte This is my paper
classification math-ph cond-mat.stat-mechmath.CAmath.DGmath.MPmath.PR MSC 34M5533E1760K3553A05 PACS 02.30.Ik33.e1702.50.Cw02.40.Hw
keywords persistence probabilityfirst-passage processesFredholm determinantsPfaffian point processessech kernelPainlevé VI equationBonnet surfacesIsing model
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper claims that the stationary persistence probability of a never-flipping spin in one-dimensional zero-temperature Ising/Potts coarsening is not just characterized by an exponent but is exactly a Fredholm Pfaffian determinant built from the integrable sech kernel, with even and odd parity sectors contributing separately. The logarithm of this determinant is controlled by a unique global solution H(x;m) of a second-order ODE identified as a Painlevé VI equation. The same function is the mean curvature of a Bonnet surface, and a quadratic folding transformation reduces the symmetric Ising case to the Manin Painlevé VI with coefficients [0,0,0,0]. From this structure the paper derives the full law's decay rate, recovering the Derrida–Hakim–Pasquier exponent for all q. If correct, the persistence law becomes a computable transcendental function rather than an asymptotic fit.

Core claim

The paper's central claim is that, for a semi-infinite one-dimensional Ising chain starting from random magnetization m and evolving by zero-temperature Glauber dynamics, the probability that the origin spin remains + for the whole stationary interval [0,ℓ] is exactly P+_0(ℓ;m) = (1+m)/2 · (D+ + D−)/2 + (1−m)/2 · (D+ − D−)/2, where D± are the even and odd Fredholm determinants of the thinned sech kernel K_sech(x)=1/(2π cosh(x/2)) with thinning parameter ξ=1−m². Each determinant equals exp(∫_0^ℓ dx (H ∓ √(−H′))/2), with H satisfying the ODE (H″/(2H′) + coth x)² + (1/sinh²x)(H²/H′ + 2H coth x + H′) = 1/4 and Cauchy data H(0+)=−ξ, H′(0+)=−H²(0+). The unique negative decreasing global solution e

What carries the argument

The central object is the translation-invariant sech kernel K_sech(x−y)=1/(2π cosh((x−y)/2)), whose even and odd restrictions to [0,ℓ] define the Fredholm determinants D±. The bridge to differential equations is the pair of resolvent kernel functions R(x,x) and R(−x,x), linked by Gaudin's relation R′=2S²; eliminating S yields the second-order second-degree ODE for H=2R. The geometric machinery identifies this ODE with the Hazzidakis first integral of Bonnet's equation, so that H is the mean curvature of a one-parameter family of Bonnet surfaces. A quadratic folding transformation of Painlevé VI, together with Borodin–Okounkov and Widom asymptotics for the determinants, fixes the connection c

Load-bearing premise

Everything probabilistic rests on the Pfaffian parity decomposition (A1)/(1.7), which is carried over from an unpublished manuscript; the paper sketches its derivation but does not fully prove it.

What would settle it

Take ℓ=4 and m=0, solve the ODE (1.9) with H(0)=−1 to high precision, compute exp(∫(H−√(−H′))/2), and compare it with a high-accuracy discretization of Det(Id−K_sech⁺) on [0,4]; any discrepancy beyond quadrature error would falsify the determinant-to-ODE link. A fully probabilistic check would measure P0(ℓ;0) by kinetic Monte Carlo on a long semi-infinite chain with t2/t1=e^ℓ and compare the whole curve, not just its slope, with the predicted determinant.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • All q-state Potts persistence exponents follow from the single function κ(2/q−1), turning the original DHP formula into a boundary value of one Painlevé VI solution.
  • The full distribution P+_0(ℓ;m) can be evaluated for arbitrary ℓ by solving one ODE or by direct Fredholm determinant computation, giving non-asymptotic predictions for simulations and experiments.
  • The same Painlevé VI system governs the first-passage probability of the stationary Gaussian process with correlator sech(T2−T1) starting from zero, so the result transfers to any system sharing that correlator.
  • The persistence exponent acquires geometric meaning: it is minus the asymptotic mean curvature at the unique umbilic point, and the Willmore energy of the Bonnet surface encodes the magnetization dependence.
  • The connection problem supplies exponentially small corrections (a Widom–Dyson constant), so the large-ℓ asymptotic expansion of the persistence probability is known beyond the leading exponent.

Where Pith is reading between the lines

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

  • If the Pfaffian identity survives scrutiny, the same full-law Painlevé VI description should extend to non-integer q through m=2/q−1, giving a one-parameter family of non-classical persistence laws that the paper does not itself test.
  • The geometric picture suggests a variational reading: the persistence law may be characterized as stationary for a Willmore-type functional on the associated Bonnet surface; the paper's remarks point this way but do not develop it into a proof.
  • For first-passage problems sharing the sech correlator, the full survival probability rather than just the 3/16 exponent should be the same Bonnet–Manin Painlevé VI; checking this at finite ℓ numerically would test universality beyond the exponent.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper claims to determine the full persistence probability distribution for the one-dimensional Ising/Potts model in the stationary scaling regime. The main result is that the persistence probability equals an even Fredholm determinant of the thinned sech kernel, expressible through a unique solution of a second-order ODE of Painlevé VI type. The persistence exponent is recovered via a Borodin-Okounkov/Wiener-Hopf asymptotic analysis, and the solution is interpreted geometrically as the mean curvature of a Bonnet surface. The paper thus proposes a universal connection between persistence in coarsening dynamics, Fredholm Pfaffians, Painlevé VI, and classical differential geometry.

Significance. If correct, the result would elevate the persistence problem from an exponent-only result (DHP, Poplavskyi–Schehr) to a full distribution function governed by a Painlevé VI transcendent, with a parameter-free connection problem. The derivation of the exponent via the Borodin-Okounkov formula is explicit and does not rely on the PVI connection problem, which lends independent support to the asymptotics. The geometric Bonnet-surface interpretation is elegant and provides a new avatar of Painlevé VI. However, the probabilistic bridge from the spin persistence probability to the even/odd Fredholm determinants is not proved in the paper, and there is a normalization inconsistency in the central Cauchy problem that must be resolved.

major comments (3)
  1. [Appendix A, Eq. (1.7)/(A1)] The Pfaffian parity decomposition is the sole probabilistic link between the Ising/Potts persistence probability and the Fredholm determinants D±. The proof is not given; the text states it was 'sketched in the unpublished work [45] by the first author.' Since every subsequent statement—Eq. (1.2), the ODE (1.9), the connection result (1.4), and the Bonnet-surface interpretation—depends on this conversion, the central claim is conditional. The authors must include a complete, self-contained proof of (A1), especially since they assert it is valid for any even difference kernel.
  2. [Section 1.1, Eq. (1.9) vs. Eq. (2.63) and small-ℓ behavior] The initial condition H(0+) = -ξ in the Cauchy problem (1.9) is inconsistent with the Fredholm determinant representation (1.2) and with the paper's own relation (2.63). For the sech kernel (1.3), K(0)=1/(2π), so the resolvent R(0+) = ξ/(2π); (2.63) then gives H(0+) = -2R(0+) = -ξ/π. The small-ℓ expansion of (1.2) yields d/dℓ log P0(0+) = (H(0+) - √(-H'(0+)))/2 = -ξ/π when H(0+) = -ξ/π and H'(0+) = -ξ²/π², matching the Neumann expansion of Det(Id−ξK+_sech). The stated condition H(0+) = -ξ gives -ξ instead, off by a factor π. This is not a notational issue; it changes the solution H and hence the distribution (1.2). The Cauchy problem (1.9) must be corrected or an explicit rescaling must be supplied.
  3. [Theorem 1.1, Eq. (1.9)] The theorem asserts without proof the existence and uniqueness of a global negative decreasing solution to the ODE (1.9). While this may follow from standard integrable-operator theory and the Fredholm determinant representation, the manuscript should either provide a proof or a precise reference, because the unique solution is the object that defines the persistence distribution P0 in (1.2).
minor comments (5)
  1. [General] The first page states 'Accepted for publication in J. Stat. Phys. (2026)' and thanks referees, which is unusual for an arXiv preprint and may be premature; this should be removed or clarified.
  2. [Section 1.3] Typo: 'ordinarry' should be 'ordinary'.
  3. [Section 2.1] Typo: 'coming from from a family' should be 'coming from a family'.
  4. [Appendix A] The paper relies on the unpublished reference [45] for the central Pfaffian identity; this should be avoided, and the proof should be included in the paper.
  5. [Appendix D, Eq. (D23)] The formula for fθ,φ(ℓ) and the subsequent Widom-Dyson constant contain Gamma-function ratios that diverge as φ→1; the manuscript handles this via Widom's theorem, but the presentation would benefit from an explicit statement of the limiting procedure for the singular case.

Circularity Check

2 steps flagged

The only load-bearing circularity is the unproved, self-cited Pfaffian decomposition (A1)/(1.7): the spin-persistence-to-Fredholm bridge is delegated to unpublished work by the first author. The subsequent Painlevé VI and Bonnet-surface analysis is independent and parameter-free.

specific steps
  1. self citation load bearing [Appendix A; identity (A1) enters as Theorem 1.1, Eq. (1.7)]
    "The proof of the identity (A1) was sketched in the unpublished work [45] by the first author. It just relies on applying the by now standard Tracy-Widom technique [115] developed to study the orthogonal and symplectic ensembles of random matrix theory, which recasts a matrix Pfaffian Fredholm determinant in terms of two scalar functions linked by the Gaudin relation (2.50)."

    Identity (A1) is the sole probabilistic bridge from the actual spin persistence probability P+_0(ℓ;m) to the even/odd Fredholm determinants D± of the thinned sech kernel. All subsequent results — the ODE (1.9), the Painlevé VI characterization, and the Bonnet-surface interpretation — are statements about these determinants. Since the proof of (A1) is only sketched in the first author's unpublished note [45] and is not reproduced here, the paper's central probabilistic claim rests entirely on a load-bearing self-citation rather than on a derivation contained in the paper or an externally verified source.

  2. self citation load bearing [Section 1.2, 'Relation to existing literature']
    "Finally, in an unpublished manuscript [45] motivated by the results of [97], the first author observed that the Ising persistence probability admits a Fredholm determinant representation involving the integrable sech kernel. However, neither the solution of the associated Painlevé VI connection problem nor the full underlying geometric structure in terms of intertwined P VI were identified at that stage."

    This passage confirms that the foundational representation of the persistence probability as a Fredholm determinant was not established in the present paper and originates in the first author's unpublished work. The 'key advance' of identifying the persistence law with a Pfaffian gap-spacing probability inherits its probabilistic validity from that self-citation. The Painlevé VI and Bonnet analysis is an independent development of the Fredholm-determinant side, but it cannot by itself supply the missing step that makes those determinants equal to the spin persistence probability.

full rationale

No fitted parameter is renamed as a prediction: the persistence exponent emerges from the Borodin-Okounkov/Wiener-Hopf asymptotics of the Fredholm determinant, and the PVI connection problem is not used to set the decay constant. The Painlevé VI classification relies on standard external results (Tracy-Widom, Okamoto, Kitaev, Manin, etc.), not on a self-citation chain that enforces the conclusion. The only load-bearing defect is the unproved, self-cited identity (A1)/(1.7) that converts the physical persistence probability into the even/odd Fredholm determinants. This is a genuine partial dependency — if (A1) fails, the 'universal persistence distribution' becomes a statement about a Fredholm determinant rather than about Ising-Potts persistence — but it is not a definitional or constructional circularity. The remaining derivation is self-contained, so the overall circularity score is moderate rather than severe.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

No free parameters beyond the physical magnetization ξ; no fitted constants and no new entities. The named 'Bonnet-Manin PVI' is a relabeling of a known equation with parameters [0,0,0,0], not a new theoretical object. The central numerics come from the Borodin-Okounkov formula, not from fitting.

axioms (5)
  • standard math Tracy-Widom / Its-Izergin-Korepin-Slavnov integrable operator machinery: the resolvent of an integrable kernel obeys a closed nonlinear ODE system
    Used in Section 2.2 (Eqs. (2.41)-(2.56)) to derive the ODE (1.9) for the sech-kernel resolvent.
  • domain assumption Pfaffian parity decomposition identity (A1)/(1.7) equating Ising/Potts persistence probability to even/odd Fredholm determinants of the thinned sech kernel
    Central probabilistic bridge; proof only sketched and attributed to unpublished [45] in Appendix A and Section 1.2.
  • standard math Bobenko-Eitner result: the mean curvature of Bonnet surfaces is a PVI tau-function (Proposition B.1, Eqs. (B20)-(B24))
    Prior literature [11,13] invoked as the basis for the geometric identification in Theorem 1.2.
  • standard math Painlevé classification of second-order second-degree ODEs (Bureau, Cosgrove, Chazy): the ODEs (2.54)-(2.56) are identified as particular PVI/CVI equations
    Section 2.3: 'one selects in the existing classifications... the very few possible matches'.
  • standard math Borodin-Okounkov formula and Widom's Fisher-Hartwig asymptotics for Wiener-Hopf determinants
    Appendix D used to solve the connection problem and obtain the limiting exponent (D2) and the ξ=1 case (D36).

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Cite this review

Pith. "Pith review of Geometry of the Ising persistence problem and the universal Bonnet-Manin Painlev\'e VI distribution." pith.science (2026). https://pith.science/paper/XQXYX4Y5

@misc{pith2026260328632,
  author       = {Pith},
  title        = {Pith review of: Geometry of the Ising persistence problem and the universal Bonnet-Manin Painlev\'e VI distribution},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XQXYX4Y5}},
  note         = {Machine review of arXiv:2603.28632}
}
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read the original abstract

We determine the full persistence probability distribution for a non-Markovian stochastic process, motivated by first-passage questions arising in interacting spin systems and allied systems. We show that this distribution is governed by a distinguished Painlev\'e VI system arising from an exact Fredholm Pfaffian structure associated with the integrable sech kernel, $K_{\mathrm{sech}}=1/(2 \pi \cosh[(x-y)/2])$. The universal persistence exponent originally obtained by Derrida, Hakim and Pasquier is recovered as an asymptotic observable and acquires a natural geometric interpretation. In the stationary scaling regime, the persistence probability admits an exact Pfaffian decomposition into even and odd Fredholm determinants of the integrable \emph{sech} kernel. These determinants are controlled by a unique global solution of a second-order nonlinear ordinary differential equation, which is identified as a particular Painlev\'e VI equation. The corresponding Painlev\'e VI connection problem determines the persistence exponent as a limiting value at infinity. We further show that the Painlev\'e VI system governing persistence admits a direct geometric interpretation: the relevant solution coincides with the mean curvature of a one-parameter family of Bonnet surfaces immersed in $\mathbb R^3$. A folding transformation between such surfaces singles out the Painlev\'e VI equation with Manin coefficients $[0,0,0,0]$, which in particular governs the universal persistence distribution in the symmetric Ising case. In this framework, the persistence exponent is identified with the asymptotic mean curvature of the associated surface.

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Reference graph

Works this paper leans on

120 extracted references · 48 canonical work pages · 1 internal anchor

  1. [1]

    Aurzada F., Simon T.,Persistence probabilities and exponents, inL´ evy matters V, Springer,183(2015)

    2015

  2. [2]

    Basor E., B¨ ottcher A., Ehrhardt T., Tracy C. A. (eds.),Toeplitz operators and random matrices, In memory of Harold Widom, Operator theory advances and ap- plications, Vol. 289, Birkh¨ auser (2022). https://doi.org/10.1007/978-3-031-13851-5 52

    work page doi:10.1007/978-3-031-13851-5 2022

  3. [3]

    Basor, E., Chen, Y.A note on Wiener-Hopf determinants and the Borodin- Okounkov identity, Integr. equ. oper. theory45, 301–308 (2003). https://doi.org/ 10.1007/s000200300006

    work page doi:10.1007/s000200300006 2003

  4. [4]

    Basor E., Widom H.,On a Toeplitz determinant identity of Borodin and Okounkov, Integr. Equ. Oper. Theory37, 397–401 (2000). https://link.springer.com/article/ 10.1007/BF01192828

    work page doi:10.1007/bf01192828 2000

  5. [5]

    L., Widom A.,Wiener-Hopf determinants with Fisher-Hartwig symbols, Operator theory advances and applications147, 131–149 (2004)

    Basor E. L., Widom A.,Wiener-Hopf determinants with Fisher-Hartwig symbols, Operator theory advances and applications147, 131–149 (2004). https://link.springer.com/chapter/10.1007/978-3-0348-7926-2_23

    work page doi:10.1007/978-3-0348-7926-2_23 2004

  6. [6]

    On the asymptotics of certain Wiener-Hopf-plus-Hankel determinants

    Basor E. L., Ehrhardt T.,On the asymptotics of certain Wiener-Hopf-plus-Hankel determinants, New York J. Math.11, 171–203 (2005). https://doi.org/10.48550/ arXiv.math/0502039 https://nyjm.albany.edu/j/2005/11-10.html

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.math/0502039 2005

  7. [7]

    High Energ

    Belitsky, A.V., Korchemsky, G.P.,Crossing bridges with strong Szeg¨ o limit theorem, J. High Energ. Phys. 2021, 257 (2021). https://doi.org/10.1007/JHEP04(2021)257

    work page doi:10.1007/jhep04(2021)257 2021

  8. [8]

    S.,Principles and applications of random noise theory, Robert E

    Bendat J. S.,Principles and applications of random noise theory, Robert E. Krieger Publishing Company (1977)

    1977

  9. [9]

    Rep.539, 225–284 (2014)

    B´ enichou O., Voituriez R.,From first-passage times of random walks in con- finement to geometry-controlled kinetics, Phys. Rep.539, 225–284 (2014). https: //doi.org/10.1016/j.physrep.2014.02.003

    work page doi:10.1016/j.physrep.2014.02.003 2014

  10. [10]

    Old and new integrable cases, 83–128, inHarmonic maps and integrable systems, Fordy A.P

    Bobenko A.I.,Surfaces in terms of 2 by 2 matrices. Old and new integrable cases, 83–128, inHarmonic maps and integrable systems, Fordy A.P. and Wood J.C. (eds.), Aspects of mathematics E23, Vieweg, Braunschweig/Wiesbaden, 1994. https://doi.org/10.1007/978-3-663-14092-4_5

    work page doi:10.1007/978-3-663-14092-4_5 1994

  11. [11]

    I., Eitner U.,Bonnet surfaces and Painlev´ e equations, Journal f¨ ur die reine und angewandte Mathematik49947–79 (1998)

    Bobenko A. I., Eitner U.,Bonnet surfaces and Painlev´ e equations, Journal f¨ ur die reine und angewandte Mathematik49947–79 (1998). https://doi.org/10.1515/ crll.1998.061

    1998

  12. [12]

    I., Kitaev A

    Bobenko A. I., Kitaev A. V.,On asymptotic cones of surfaces with constant curvature and the third Painlev´ e equation, Manuscripta math.97489-516 (1998). https://doi.org/10.1007/s002290050117

    work page doi:10.1007/s002290050117 1998

  13. [13]

    I., Eitner U.,Painlev´ e equations in differential geometry of surfaces, Lecture notes in math.1753(Springer, Berlin, 2000)

    Bobenko A. I., Eitner U.,Painlev´ e equations in differential geometry of surfaces, Lecture notes in math.1753(Springer, Berlin, 2000). https://doi.org/10.1007/ b76883

    2000

  14. [14]

    I.,Exploring surfaces through methods from the Theory of integrable systems: The Bonnet problem, In: Surveys on geometry and integrable systems, Eds

    Bobenko A. I.,Exploring surfaces through methods from the Theory of integrable systems: The Bonnet problem, In: Surveys on geometry and integrable systems, Eds. Guest M., Miyaoka R., Ohnita Y. Advanced studies in pure mathematics 51, 1–51 (2008). https://projecteuclid.org/ebook/Download?urlid=10.2969/aspm/ 53 05110001&isFullBook=false http://www.arxiv.org...

    Pith/arXiv arXiv 2008

  15. [15]

    Deuxi` eme partie: D´ etermination de toutes les surfaces applicables sur une surface donn´ ee, J.´Ec

    Bonnet P.-O.,M´ emoire sur la th´ eorie des surfaces applicables sur une surface donn´ ee. Deuxi` eme partie: D´ etermination de toutes les surfaces applicables sur une surface donn´ ee, J.´Ec. Polytech.42, 1–151 (1867). http://gallica.bnf.fr/ark:/12148/ bpt6k433698b/f5.image

  16. [16]

    Pure Appl

    Borodin A., Deift P.,Fredholm determinants, Jimbo-Miwa-Uenoτ-functions, and representation theory, Comm. Pure Appl. Math.55, 1160–1230 (2002). https://doi. org/10.1002/cpa.10042 https://arxiv.org/abs/math-ph/0111007

    Pith/arXiv arXiv 2002

  17. [17]

    Borodin A., Okounkov A.,A Fredholm determinant formula for Toeplitz de- terminants, Int. Eq. Oper. Theory37, 386–396 (2000). https://doi.org/10.1007/ BF01192827 https://arxiv.org/abs/math/9907165

    Pith/arXiv arXiv 2000

  18. [18]

    Determinantal ensembles, J

    Borodin, A.; Soshnikov, A.,Janossy densities I. Determinantal ensembles, J. Stat. Phys.113, 595–610 (2003). https://doi.org/10.1023/A:1026025003309 https: //arxiv.org/abs/math-ph/0212063

    Pith/arXiv arXiv 2003

  19. [19]

    Math 345, 483–551 (2019)

    Bothner T., Its A., Prokhorov A.,On the analysis of incomplete spectra in random matrix theory through an extension of the Jimbo-Miwa-Ueno differential, Adv. Math 345, 483–551 (2019). https://doi.org/10.1016/j.aim.2019.01.025

    work page doi:10.1016/j.aim.2019.01.025 2019

  20. [20]

    J.,Theory of phase ordering kinetics, Adv

    Bray A. J.,Theory of phase ordering kinetics, Adv. Phys.43, 357 (1994). https: //doi.org/10.1080/00018730110117433

    doi:10.1080/00018730110117433 1994

  21. [21]

    J., Majumdar S

    Bray A. J., Majumdar S. N., Schehr G.,Persistence and first-passage properties in non-equilibrium systems, Adv. Phys.62, 225 (2013). https://doi.org/10.1080/ 00018732.2013.803819

    arXiv 2013

  22. [22]

    Bufetov A.,The expectation of a multiplicative functional under the sine- process, Funct. Anal. Appl.58(2024), 23—33 (2024). https://doi.org/10.1134/ S0016266324020035

    2024

  23. [23]

    https://doi.org/10.1007/BF02428819

    Bureau F.J., ´Equations diff´ erentielles du second ordre enYet du second degr´ e en ¨Ydont l’int´ egrale g´ en´ erale est ` a points critiques fixes, Annali di matematica pura ed applicata91, 1–281 (1972). https://doi.org/10.1007/BF02428819

    work page doi:10.1007/bf02428819 1972

  24. [24]

    B.,The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell, J

    Canham P. B.,The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell, J. Theor. Biol.26, 61—81 (1970). https://doi.org/10.1016/S0022-5193(70)80032-7

    doi:10.1016/s0022-5193(70)80032-7 1970

  25. [25]

    Cartan ´E.,Sur les couples de surfaces applicables avec conservation des courbures principales, Bull. Sci. Math.66, 55–72; 74–85 (1942). https://gallica.bnf.fr/ark: /12148/bpt6k9632895z.r=Cartan?rk=42918;4 54

    1942

  26. [26]

    Math.34, 317–385 (1911)

    Chazy J.,Sur les ´ equations diff´ erentielles du troisi` eme ordre et d’ordre sup´ erieur dont l’int´ egrale g´ en´ erale a ses points critiques fixes, Acta. Math.34, 317–385 (1911). https://dx.doi.org/10.1007/BF02393131

    work page doi:10.1007/bf02393131 1911

  27. [27]

    Chern S.S.,Deformation of surfaces preserving principal curvatures, 155–163, InDifferential geometry and complex analysis, eds. I. Chavel and H.M. Farkas, H.E. Rauch memorial volume (Springer, Berlin, 1985). https://doi.org/10.1007/978-3-642-69828-6_10

    work page doi:10.1007/978-3-642-69828-6_10 1985

  28. [28]

    Conte R.,On the Schlesinger transformations of the sixth Painlev´ e equation, C. R. Math. Acad. Sci. Paris,332, 501–504 (2001). http://dx.doi.org/10.1016/ S0764-4442(01)01944-9 http://arXiv.org/abs/math.CA/0103165

    arXiv 2001

  29. [29]

    Musette M.,New contiguity relation of the sixth Painlev´ e equation from a truncation, Physica D161, 129–141 (2002)

    Conte R., M. Musette M.,New contiguity relation of the sixth Painlev´ e equation from a truncation, Physica D161, 129–141 (2002). http://dx.doi.org/10.1016/ S0167-2789(01)00372-4 http://arXiv.org/abs/nlin.SI/0112007

    arXiv 2002

  30. [30]

    http://www.kurims.kyoto-u.ac.jp/ ∼kenkyubu/bessatsu/open/B2/pdf/B2-3.pdf https://arxiv.org/abs/nlin/0701049

    Conte R.,On the Lax pairs of the sixth Painlev´ e equation, RIMS Kˆ okyˆ uroku BessatsuB2, 21–27 (2007). http://www.kurims.kyoto-u.ac.jp/ ∼kenkyubu/bessatsu/open/B2/pdf/B2-3.pdf https://arxiv.org/abs/nlin/0701049

    Pith/arXiv arXiv 2007

  31. [31]

    Math.,137, 306–327 (2016)

    Conte R., Grundland A.M.,Reductions of Gauss-Codazzi equations, Studies in Appl. Math.,137, 306–327 (2016). http://dx.doi.org/10.1111/sapm.12121 http:// arXiv.org/abs/1601.04300

    Pith/arXiv arXiv 2016

  32. [32]

    Conte R.,Surfaces de Bonnet et ´ equations de Painlev´ e, C. R. Math. Acad. Sci. Paris355, 40–44 (2017). https://doi.org/10.1016/j.crma.2016.10.019 http://arXiv. org/abs/1607.01222v2

    Pith/arXiv arXiv 2017

  33. [33]

    Conte R.,Generalized Bonnet surfaces and Lax pairs ofP V I, J. Math. Phys.58, 103508 (2017). https://doi.org/10.10/1.4995689 http://arXiv.org/abs/1710.04944

    Pith/arXiv arXiv 2017

  34. [34]

    https://doi.org/10.1007/978-3-030-53340-3

    Conte R., Musette M.,The Painlev´ e handbook(second edition), (Springer Nature, Switzerland, 2020). https://doi.org/10.1007/978-3-030-53340-3

    doi:10.1007/978-3-030-53340-3 2020

  35. [35]

    M., Scoufis A.,Painlev´ e classification of a class of differential equations of the second order and second degree, Stud

    Cosgrove C. M., Scoufis A.,Painlev´ e classification of a class of differential equations of the second order and second degree, Stud. Appl. Math.88, 25–87 (1993). https://doi.org/10.1002/sapm199388125

    work page doi:10.1002/sapm199388125 1993

  36. [36]

    M.,Chazy’s second-degree Painlev´ e equations, J

    Cosgrove C. M.,Chazy’s second-degree Painlev´ e equations, J. Phys. A39, 11955– 11971 (2006). https://doi.org/10.1088/0305-4470/39/39/S01

    work page doi:10.1088/0305-4470/39/39/s01 2006

  37. [37]

    Pure Appl

    Deift P., Its A., Krasovsky I,.Toeplitz matrices and Toeplitz determinants under the impetus of the Ising model: Some history and some recent results, Comm. Pure Appl. Math.66, 1360–1438 (2013). https://arxiv.org/abs/1207.4990 55

    Pith/arXiv arXiv 2013

  38. [38]

    M., Zeitouni O.,Random polynomials having few or no real zeros, J

    Dembo A., Poonen B., Shao Q. M., Zeitouni O.,Random polynomials having few or no real zeros, J. Amer. Math. Soc.15, 857 (2002). https://doi.org/10.1090/ S0894-0347-02-00386-7

    2002

  39. [39]

    Probab.43, 85 (2015)

    Dembo A., Mukherjee S.,No zero-crossings for random polynomials and the heat equation, Ann. Probab.43, 85 (2015). https://doi.org/10.1214/13-AOP852

    work page doi:10.1214/13-aop852 2015

  40. [40]

    Derrida B., Bray A. J. and Godr` eche C.,Nontrivial exponents in the zero- temperature dynamics of the 1d Ising and Potts models, J. Phys. A.27, L357 (1994). https://doi.org/10.1088/0305-4470/27/11/002

    work page doi:10.1088/0305-4470/27/11/002 1994

  41. [41]

    Derrida B., Hakim V., Pasquier V.,Exact first-passage exponents of 1d domain growth—relation to a reaction-diffusion model, Phys. Rev. Lett.75, 751 (1995). https://doi.org/10.1103/PhysRevLett.75.751

    work page doi:10.1103/physrevlett.75.751 1995

  42. [42]

    Derrida B., Hakim V., Pasquier V.,Exact exponent for the number of persistent spins in the zero-temperature dynamics of the one-dimensional Potts model, J. Stat. Phys.85, 7 (1996). https://doi.org/10.1007/BF02199362

    work page doi:10.1007/bf02199362 1996

  43. [43]

    Derrida B., Hakim V., Zeitak R.,Persistent spins in the linear diffusion approx- imation of phase ordering and zeros of stationary gaussian processes, Phys. Rev. Lett.77, 2871 (1996). https://doi.org/10.1103/PhysRevLett.77.2871

    work page doi:10.1103/physrevlett.77.2871 1996

  44. [44]

    J.,Relationships betweenτ-functions and Fredholm determinant expressions for gap probabilities in random matrix theory, Nonlinearity 19, 1643—1656 (2006)

    Desrosiers P., Forrester P. J.,Relationships betweenτ-functions and Fredholm determinant expressions for gap probabilities in random matrix theory, Nonlinearity 19, 1643—1656 (2006). https://doi.org/10.1088/0951-7715/19/7/012

    work page doi:10.1088/0951-7715/19/7/012 2006

  45. [45]

    06957 (unpublished)

    Dornic I,Universal Painlev´ e VI probability distribution in Pfaffian persistence and Gaussian first-passage problems with a sech kernel, https://arxiv.org/abs/1810. 06957 (unpublished)

  46. [46]

    J.,The threefold way

    Dyson F. J.,The threefold way. Algebraic structure of symmetry groups and ensembles in quantum mechanics, J. Math. Phys.3(6)1199–1215 (1962). https: //doi.org/10.1063/1.1703863

    doi:10.1063/1.1703863 1962

  47. [47]

    J.,Fredholm determinants and inverse scattering problems, Com- mun

    Dyson F. J.,Fredholm determinants and inverse scattering problems, Com- mun. Math. Phys.47, 171–183 (1976). https://doi.org/10.1007/BF01608375

    work page doi:10.1007/bf01608375 1976

  48. [48]

    Edelman A., Kostlan E.,How many zeros of a random polynomial are real?, Bull. Amer. Math. Soc.32, 1–37 (1995). https://www.ams.org/journals/bull/ 1995-32-01/S0273-0979-1995-00571-9/

    1995

  49. [49]

    E., Hartwig R

    Fisher M. E., Hartwig R. E.,Toeplitz determinants: some applications, theo- rems, and conjectures, Adv. Chem. Phys.15, 333 (1968). https://doi.org/10.1002/ 9780470143605.ch18 56

    1968

  50. [50]

    Probab.50, 2409–2474 (2022)

    Fitzgerald W., Tribe R., Zaboronski O.,Asymptotic expansions for a class of Fredholm Pfaffians and interacting particle systems, Ann. Probab.50, 2409–2474 (2022). https://doi.org/10.1214/22-AOP1586

    work page doi:10.1214/22-aop1586 2022

  51. [51]

    J., Witte N

    Forrester P. J., Witte N. S.,Application of theτ-function theory of Painlev´ e equations to random matrices:P VI, the JUE, CyUE, and scaled limits, Nagoya Math. J.174, 29–114 (2004). https://www.osti.gov/etdeweb/biblio/20194838

    arXiv 2004

  52. [52]

    J.,The limiting Kac random polynomial and truncated ran- dom orthogonal matrices, J

    Forrester P. J.,The limiting Kac random polynomial and truncated ran- dom orthogonal matrices, J. Stat. Mech. P12018 (2010). https://doi.org/10.1088/ 1742-5468/2010/12/P12018

    2010

  53. [53]

    J.,Diffusion processes and the asymptotic bulk gap probability for the real Ginibre ensemble, J

    Forrester P. J.,Diffusion processes and the asymptotic bulk gap probability for the real Ginibre ensemble, J. Phys. A: Math. Theor.48324001 (2015). https://doi. org/10.1088/1751-8113/48/32/324001 https://arxiv.org/abs/1306.4106

    Pith/arXiv arXiv 2015

  54. [54]

    Fuchs R.,Sur quelques ´ equations diff´ erentielles lin´ eaires du second ordre, C.R. Math. Acad. Sci. Paris141555–558 (1905). http://refhub.elsevier.com/ S1631-073X(14)00173-3/bib46756368735036s1

    1905

  55. [55]

    Fuchs R., ¨Uber lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegenen wesentlich singul¨ aren Stellen, Math. Ann. , 301–321 (1907). https://doi.org/10.1007/BF01449199

    work page doi:10.1007/bf01449199 1907

  56. [56]

    https://doi.org/10.1007/JHEP10(2012) 038

    Gamayun O., Iorgov N., Lisovyy O.,Conformal field theory of Painlev´ e VI, Jour- nal of high energy physics10, 38 (2012). https://doi.org/10.1007/JHEP10(2012) 038

    doi:10.1007/jhep10(2012 2012

  57. [57]

    Gamayun O., Iorgov N., Lisovyy O.,How instanton combinatorics solves Painlev´ e VI,V and III’s, J. Phys. A: Math. Theor.46, 335203 (2013). https://dx.doi.org/10. 1007/JHEP10(2012)038 Erratum, https://dx.doi.org/10.1007/JHEP10(2012)183 http://arXiv.org/abs/1207.0787v3

    Pith/arXiv arXiv 2013

  58. [58]

    Phys.25, 447–458 (1961)

    Gaudin M.,Sur la loi limite de l’espacement des valeurs propres d’une matrice al´ eatoire, Nucl. Phys.25, 447–458 (1961). https://doi.org/10.1016/0029-5582(61) 90176-6

    work page doi:10.1016/0029-5582(61 1961

  59. [59]

    Geronimo J.S., Case K.M.,Scattering theory and polynomials orthogonal on the unit circle, J. Math. Phys.20, 299 (1979). https://doi.org/10.1063/1.524077

    work page doi:10.1063/1.524077 1979

  60. [60]

    and Semengue J-P.,The Jacobi polynomial ensemble and the Painlev´ e VI, J

    Haine L. and Semengue J-P.,The Jacobi polynomial ensemble and the Painlev´ e VI, J. Math. Phys.402117–2134 (1999). https://doi.org/10.1063/1.532855

    work page doi:10.1063/1.532855 1999

  61. [61]

    P., McLeod J

    Hastings S. P., McLeod J. B.,A boundary value problem associated with the second Painlev´ e transcendent and the Korteweg-de Vries equation, Arch. Rat. Mech. Anal., 73, 31–51 (1980). https://doi.org/10.1007/BF00283254 57

    doi:10.1007/bf00283254 1980

  62. [62]

    N,Biegung mit Erhaltung der Hauptkr¨ ummungsradien, J

    Hazzidakis J. N,Biegung mit Erhaltung der Hauptkr¨ ummungsradien, J. Reine Angew. Math.117, 42–56 (1897). https://doi.org/10.1515/crll.1897.117.42

    work page doi:10.1515/crll.1897.117.42

  63. [63]

    Helfrich W.,Elastic properties of lipid bilayers: Theory and possible experiments. Z. Naturforsch.28, 693–703 (1973). https://cyberleninka.org/article/n/881934.pdf

    1973

  64. [64]

    R., Izergin A

    Its A. R., Izergin A. G., Korepin V. E., Slavnov N. A.,Differential equations for quantum correlation functions, Int. J. Mod. Phys. B4, 1003–1037 (1990). https: //doi.org/10.1142/S0217979290000504

    doi:10.1142/s0217979290000504 1990

  65. [65]

    https://doi

    Jimbo M., Miwa T., Mˆ ori Y., Sato M.,Density matrix of an impenetrable Bose gas and the fifth Painlev´ e transcendent, Physica D1, 80–158 (1980). https://doi. org/10.1016/0167-2789(80)90006-8

    work page doi:10.1016/0167-2789(80)90006-8 1980

  66. [66]

    XVII, Proc

    Jimbo M., Miwa T.,Studies on holonomic quantum fields. XVII, Proc. Japan Acad.56405–410 (1980). https://doi.org/10.3792/pjaa.56.405

    work page doi:10.3792/pjaa.56.405 1980

  67. [67]

    II, Physica D2, 407–448 (1981)

    Jimbo M., Miwa T.,Monodromy preserving deformations of linear ordinary differential equations with rational coefficients. II, Physica D2, 407–448 (1981). https://doi.org/10.1016/0167-2789(81)90021-X

    work page doi:10.1016/0167-2789(81)90021-x 1981

  68. [68]

    RIMS, Kyoto University,18, 1137–1161 (1982)

    Jimbo M.,Monodromy problem and the boundary condition for some Painlev´ e transcendents, Publ. RIMS, Kyoto University,18, 1137–1161 (1982). https://doi. org/10.2977/prims/1195183300

    arXiv 1982

  69. [69]

    Kabluchko Z.,An infinite-dimensional helix invariant under spherical projections, Electron. Commun. Probab.24, 1–13 (2019). https://doi.org/10.1214/19-ECP234

    work page doi:10.1214/19-ecp234 2019

  70. [70]

    Kac M.,On the average number of real roots of a random algebraic equation, Bull. Amer. Math. Soc.49314–320 (1943) https://projecteuclid. org/journals/bulletin-of-the-american-mathematical-society/volume-49/issue-4/ On-the-average-number-of-real-roots-of-a-random/bams/1183505112.pdf

    arXiv 1943

  71. [71]

    J.21, 504 (1954)

    Kac M.,Toeplitz matrices, translation kernels, and a related problem in probability theory, Duke Math. J.21, 504 (1954). https://doi.org/10.1215/ S0012-7094-54-02149-3

    1954

  72. [72]

    Math.,96, 9–51 (1996)

    Konopelchenko B.G.,Induced surfaces and their integrable dynamics, Studies in Appl. Math.,96, 9–51 (1996). https://doi.org/10.1002/sapm19969619

    work page doi:10.1002/sapm19969619 1996

  73. [73]

    Kitaev A.V.,Quadratic transformations for the sixth Painlev´ e equation, Lett. Math. Phys.21, 105–111 (1991). https://doi.org/10.1007/BF00401643

    work page doi:10.1007/bf00401643 1991

  74. [74]

    V., Shao Q

    Li W. V., Shao Q. M.,Lower tail probabilities for Gaussian processes, Ann. Prob. 32, 216–242 (2004). https://doi.org/10.1214/aop/1078415834

    arXiv 2004

  75. [75]

    M,Interacting particle systems, Springer-Verlag Berlin-Heidelberg (1985, 2005)

    Liggett T. M,Interacting particle systems, Springer-Verlag Berlin-Heidelberg (1985, 2005). https://doi.org/10.1007/b138374 58

    doi:10.1007/b138374 1985

  76. [76]

    Lisovyy O., Tykhyy Y.,Algebraic solutions of the sixth Painlev´ e equation, J. Geom. Phys.85, 124–1 (2014). https://doi.org/10.1016/j.geomphys.2014.05.010 https://arXiv.org/abs/0809.4873v2

    Pith/arXiv arXiv 2014

  77. [77]

    Math.80, 113–166 (2016)

    Loray F.,Isomonodromic deformation of Lam´ e connections, Painlev´ e VI equation and Okamoto symmetry, Izv. Math.80, 113–166 (2016). https://doi.org/10.1070/ IM8310 http://arXiv.org/abs/1410.4976

    Pith/arXiv arXiv 2016

  78. [78]

    N., Sire C., Bray A

    Majumdar S. N., Sire C., Bray A. J. and Cornell S. J.,Nontrivial exponent for simple diffusion, Phys. Rev. Lett.77, 2867–2870 (1996). https://doi.org/10.1103/ PhysRevLett.77.2867

    1996

  79. [79]

    N.,Persistence in nonequilibrium systems, Curr

    Majumdar S. N.,Persistence in nonequilibrium systems, Curr. Sci. (India)77, 370 (1999). https://www.jstor.org/stable/24102955

    arXiv 1999

  80. [80]

    Malmquist J.,Sur les ´ equations diff´ erentielles du second ordre, dont l’int´ egrale g´ en´ erale a ses points critiques fixes, Arkiv Math. Astr. Fys.17, 1–89 (1922). https: //hdl.handle.net/2027/inu.30000100110653

    1922

Showing first 80 references.