Chaire Jean-Morlet : Equation intégrable aux données initiales aléatoires / Jean-Morlet Chair : Integrable Equation with Random Initial Data

Collection Chaire Jean-Morlet : Equation intégrable aux données initiales aléatoires / Jean-Morlet Chair : Integrable Equation with Random Initial Data

Organizer(s) Basor, Estelle ; Bufetov, Alexander ; Cafasso, Mattia ; Grava, Tamara ; McLaughlin, Ken
Date(s) 08/04/2019 - 12/04/2019
linked URL https://www.chairejeanmorlet.com/2104.html
00:00:00 / 00:00:00
19 22

A tale of Pfaffian persistence tails told by a Bonnet-Painlevé VI transcendent

By Ivan Dornic

We identify the persistence probability for the zero-temperature non-equilibrium Glauber dynamics of the half-space Ising chain as a particular Painlevé VI transcendent, with monodromy exponents (1/2,1/2,0,0). Among other things, this characterization a la Tracy-Widom permits to relate our specific Bonnet-Painlevé VI to the one found by Jimbo & Miwa and characterizing the diagonal correlation functions for the planar static Ising model. In particular, in terms of the standard critical exponents eta=1/4 and beta=1/8 for the latter, this implies that the probability that the limiting Gaussian real Kac's polynomial has no real root decays with an exponent 4(eta+beta)=3/4.

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Citation data

  • DOI 10.24350/CIRM.V.19517703
  • Cite this video Dornic, Ivan (12/04/2019). A tale of Pfaffian persistence tails told by a Bonnet-Painlevé VI transcendent. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19517703
  • URL https://dx.doi.org/10.24350/CIRM.V.19517703

Bibliography

  • B. Derrida, V. Hakim, V. Pasquier, Exact Exponent for the Number of Persistent Spins in the Zero-Temperature Dynamics of the One-Dimensional Potts Model, J. Stat. Phys. 85, 763 (1996) - https://doi.org/10.1007/BF02199362
  • C. A. Tracy, H. Widom, Fredholm determinants, differential equations, and matrix models, Commun. Math. Phys. 163, 33-72 (1994). - https://doi.org/10.1007/BF02101734
  • C. A. Tracy, H. Widom, On orthogonal and symplectic matrix ensembles, Commun. Math. Phys. 177, 727-754 (1996). - https://doi.org/10.1007/BF02099545
  • A.I. Bobenko, U. Eitner, Bonnet Surfaces and Painlevé Equations, J. Reine Angew. Math. 499, 47-79 (1998) - https://doi.org/10.1515/crll.1998.061
  • Tsuda, Okamoto, Sakai, Folding transformations of the Painlevé equations, Math. Annal. 331, 919 (2005) - https://doi.org/10.1007/s00208-004-0600-8
  • Matsumoto, Shirai, Correlation functions for zeros of a Gaussian power series and Pfaffians, Electron. J. Probab. 18, 1 (2013) - http://dx.doi.org/10.1214/EJP.v18-2545
  • Patrick Desrosiers and Peter J Forrester ; Relationships between τ-functions and Fredholm determinant expressions for gap probabilities in random matrix theory; 2006 Nonlinearity n°19 p.1643 - https://doi.org/10.1088/0951-7715/19/7/012
  • Bonnet, O. (1867). Mémoire sur la théorie des surfaces applicables sur une surface donnée. deuxième partie. Gauthier-Villars.
  • Robert Conte; Generalized Bonnet surfaces and Lax pairs of PVI Journal of Mathematical Physics 2017 58:10 - https://doi.org/10.1063/1.4995689
  • Alexander I. BobenkoUlrich Eitner ; Painlevé Equations in the Differential Geometry of Surfaces; LNM, volume 1753 ; Springer; 2000; 978-3-540-41414-8 - https://doi.org/10.1007/b76883

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