Generalized Bonnet surfaces and Lax pairs of PVI

Conte, Robert

doi:10.1063/1.4995689

Cited by 8 publications

(5 citation statements)

References 38 publications

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“…It is also hoped that their universal behavior will one day incarnates into genuine nonequilibrium physical systems, with experiments as striking as those conducted for KPZ-growing interfaces associated to the P II Tracy-Widom distributions [41][42][43]. The correspondence between a generic P VI and Bonnet surfaces established in [60,61] should also explain [90], by the classical confluence along the Painlevé hierarchy all the way down to P II , why geometry-dependent universality classes are observed in KPZ growth, where these nonequilibrium interfaces forever remembers the initial curvature they had.…”

Section: Universality Aspects and Conclusionmentioning

confidence: 95%

“…The latter remarkable correspondence uses the fact that the Gauss-Codazzi equations for the moving frame of a Bonnet surface can be retranscribed as a Lax pair for P VI [56,57]. As shown in 2017 by Robert Conte [60,61], the corresponding codimension-three Bonnet-P VI can be extrapolated to the "full" P VI with four arbitrary monodromy parameters to obtain, due to its geometric origin, probably the "best" (more symmetric) Lax pair.…”

Section: The Painlev é VI Bonnet Surfacementioning

confidence: 99%

“…( 11) with Eq. (B.20) from [61]), with ℜz = T , and the specific Bonnet-P VI , Eq. ( 1) for y = y(x), with x = e 2T , and its accompanying monodromy exponents, Eq.…”

Section: The Painlev é VI Bonnet Surfacementioning

confidence: 99%

“…with still T = ℜz, while ℑ(z) is related to the spectral parameter in the Lax pair representation of the moving frame [55,61], the piece 1/ sinh 2 T in Eq. (18 being the so-called Hopf factor of the surface.…”

Section: The Painlev é VI Bonnet Surfacementioning

confidence: 99%

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Universal Painlevé VI Probability Distribution in Pfaffian Persistence and Gaussian First-Passage Problems with a sech-Kernel

Dornic¹

2018

Preprint

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We recast the persistence probability for the spin located at the origin of a half-space arbitrarily m-magnetized Glauber-Ising chain as a Fredholm Pfaffian gap probability generating function with a sech-kernel. This is then spelled out as a tau-function for a certain Painlevé VI transcendent, the persistence exponent θ(m)/2 emerging as an asymptotic decay rate. Using a known yet remarkable correspondence that relates Painlevé equations to Bonnet surfaces, the persistence probability also acquires a geometric meaning in terms of the mean curvature of the latter, and even a topological one at the magnetization-symmetric point in terms of Gauss intrinsic curvature. Since the same sechkernel with an underlying Pfaffian structure shows up in a variety of Gaussian first-passage problems, our Painlevé VI provides their universal first-passage probability distribution, in a manner exactly analogous to the famous Painlevé II Tracy-Widom laws. The tail behavior in the magnetizationsymmetric case of our full scaling function allows to recover the exact persistence exponent θ(0)/2 = 3/16 for the 2d-diffusing random field or for random real Kac's polynomials, a particular result found very recently by Poplavskyi and Schehr [Phys. Rev. Lett. 121, 150601 (2018)]. Our Painlevé VI tau-function persistence distribution also bears a correspondence with a c = 1 conformal field theory, the monodromy parameters giving the dimensions of the associated primary fields. This yields θ(0) = 3β, with β = 1/8 the Onsager-Yang magnetization exponent for the critical 2d Ising model, a plausible conjecture relating a nonequilibrium exponent to ordinary static critical behavior in one more space dimension, that suggests more generally that methods of boundary conformal field theory should be helpful in determining the critical properties of other unsolved nonequilibrium 1d processes.

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Section: Universality Aspects and Conclusionmentioning

confidence: 95%

Section: The Painlev é VI Bonnet Surfacementioning

confidence: 99%

“…( 11) with Eq. (B.20) from [61]), with ℜz = T , and the specific Bonnet-P VI , Eq. ( 1) for y = y(x), with x = e 2T , and its accompanying monodromy exponents, Eq.…”

Section: The Painlev é VI Bonnet Surfacementioning

confidence: 99%

Section: The Painlev é VI Bonnet Surfacementioning

confidence: 99%

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Universal Painlevé VI Probability Distribution in Pfaffian Persistence and Gaussian First-Passage Problems with a sech-Kernel

Dornic¹

2018

Preprint

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“…In the classical (non-super) geometry of surfaces, Bonnet surfaces are known for their solutions linked with the Painlevé P6 equation (see e.g. [3] and references therein). It would be interesting to investigate if similar properties appear in a Bonnet-like supermanifold.…”

Section: Structural Equations For a Manifold Of Type F 3 In A Spheric...mentioning

confidence: 99%

Structural equations of supermanifolds immersed in the superspace ${\mathcal{M}^{(3\vert2)}(c)}$ with a prescribed curvature

Bertrand¹,

Grundland²

2018

J. Phys. A: Math. Theor.

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The aim of this paper is to construct the structural equations of supermanifolds immersed in Euclidean, hyperbolic and spherical superspaces parametrised with two bosonic and two fermionic variables. To perform this analysis, for each type of immersion, we split the supermanifold into its Grassmannian components and study separately each manifold generated. Even though we consider four variables in the Euclidean case, we obtain that the structural equations of each manifold are linked with the Gauss-Codazzi equations of a surface immersed in a Euclidean or spherical space. In the hyperbolic and spherical superspaces, we find that the body manifolds are linked with the classical Gauss-Codazzi equations for a surface immersed in hyperbolic and spherical spaces, respectively. For some soul manifolds, we show that the immersion of the manifolds must be in a hyperbolic space and that the structural equations split into two cases. In one case, the structural equations reduce to the Liouville equation, which can be completely solved. In the other case, we can express the geometric quantities solely in terms of the metric coefficients, which provide a geometric characterization of the structural equations in terms of functions linked with the Hopf differential, the mean curvature and a new function which does not appear in the characterization of a classical (not super) surface. Keywords: supersymetric model, supermanifold, spherical and hyperbolic immersion, structural equations of surface.Structural equations of supermanifolds immersed in M (3|2) (c)

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Selected Problems Integrated by Painlevé Functions

Conte

Musette

2020

The Painlevé Handbook

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Generalized Bonnet surfaces and Lax pairs of PVI

Cited by 8 publications

References 38 publications

Universal Painlevé VI Probability Distribution in Pfaffian Persistence and Gaussian First-Passage Problems with a sech-Kernel

Universal Painlevé VI Probability Distribution in Pfaffian Persistence and Gaussian First-Passage Problems with a sech-Kernel

Structural equations of supermanifolds immersed in the superspace ${\mathcal{M}^{(3\vert2)}(c)}$ with a prescribed curvature

Selected Problems Integrated by Painlevé Functions

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