Ir al contenido

Documat

Pole dynamics and an integral of motion for multiple SLE(0)

  • Tom Alberts [1] ; Sung-Soo Byun [2] ; Nam-Gyu Kang [3] ; Nikolai G. Makarov [4]
    1. [1] University of Utah

      University of Utah

      Estados Unidos

    2. [2] Seoul National University

      Seoul National University

      Corea del Sur

    3. [3] Korea Institute for Advanced Study

      Korea Institute for Advanced Study

      Corea del Sur

    4. [4] California Institute of Technology

      California Institute of Technology

      Estados Unidos

    Mostrar afiliaciones +
  • Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 30, Nº. 5, 2024
  • Idioma: inglés
  • DOI: 10.1007/s00029-024-00980-0
  • Enlaces
  • Resumen

    • We describe the Loewner chains of the real locus of a class of real rational functions whose critical points are on the real line. Our main result is that the poles of the rational function lead to explicit formulas for the dynamical system that governs the driving functions. Our formulas give a simple method for mapping the class of rational functions into solutions to a non-trivial system of quadratic equations, and for directly showing that the curves in the real locus satisfy geometric commutation. These results are entirely self-contained and have no reliance on probabilistic objects, but make use of an integral of motion for the Loewner chain that is motivated by ideas from conformal field theory. We also show that the dynamics of the driving functions are a special case of the Calogero–Moser integrable system, restricted to a particular submanifold of phase space carved out by the Lax matrix. Our approach complements a recent result of Peltola and Wang, who showed that the real locus is the deterministic κ→0 limit of the multiple SLE(κ) curves.

  • Referencias bibliográficas
    • 1. Abanov, A.G., Bettelheim, E.,Wiegmann, P.: Integrable hydrodynamics of Calogero-Sutherland model: bidirectional Benjamin-Ono equation....
    • 2. Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Society Lecture Note...
    • 3. Abanov, A.G., Gromov, A., Kulkarni, M.: Soliton solutions of a Calogero model in a harmonic potential. J. Phys. A Math. Theor. 44(29),...
    • 4. Alberts, T., Kang, N.-G., Makarov, N. G.: Conformal field theory for multiple SLEs, in preparation
    • 5. Airault, H., McKean, H.P., Moser, J.: Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem....
    • 6. Bauer, M., Bernard, D., Kytölä, K.: Multiple Schramm-Loewner evolutions and statistical mechanics martingales. J. Stat. Phys. 120(5–6),...
    • 7. Bonk, M., Eremenko, A.: Canonical embeddings of pairs of arcs. Comput. Methods Funct. Theory 21, 825–830 (2021)
    • 8. Benzinger, H.E.: Plane autonomous systems with rational vector fields. Trans. Am. Math. Soc. 326(2), 465–483 (1991)
    • 9. Beffara, V., Peltola, E., Hao, W.: On the uniqueness of global multiple SLEs. Ann. Probab. 49(1), 400–434 (2021)
    • 10. Calogero, F.: Solution of the one-dimensional N-body problems with quadratic and/or inversely quadratic pair potentials. J. Math. Phys....
    • 11. Cardy, J.: Calogero-Sutherland model and bulk-boundary correlations in conformal field theory. Phys. Lett. B 582(1–2), 121–126 (2004)
    • 12. Doyon, B., Cardy, J.: Calogero-Sutherland eigenfunctions with mixed boundary conditions and conformal field theory correlators. J. Phys....
    • 13. Dubédat, J.: Euler integrals for commuting SLEs. J. Stat. Phys. 123(6), 1183–1218 (2006)
    • 14. Dubédat, J.: Commutation relations for Schramm-Loewner evolutions. Comm. Pure Appl. Math. 60(12), 1792–1847 (2007)
    • 15. Dubédat, J.: SLE and the free field: partition functions and couplings. J. Amer. Math. Soc. 22(4), 995–1054 (2009)
    • 16. Eremenko, A., Gabrielov, A.: Rational functions with real critical points and the B. and M. Shapiro conjecture in real enumerative geometry....
    • 17. Eremenko, A., Gabrielov, A.: An elementary proof of the B. and M. Shapiro conjecture for rational functions, Notions of positivity and...
    • 18. Flores, S.M., Kleban, P.: A solution space for a system of null-state partial differential equations: part 1. Comm. Math. Phys. 333(1),...
    • 19. Flores, S.M., Kleban, P.: A solution space for a system of null-state partial differential equations: part 2. Comm. Math. Phys. 333(1),...
    • 20. Flores, S.M., Kleban, P.: A solution space for a system of null-state partial differential equations: part 3. Comm. Math. Phys. 333(2),...
    • 21. Flores, S.M., Kleban, P.: A solution space for a system of null-state partial differential equations: part 4. Comm. Math. Phys. 333(2),...
    • 22. Garijo, A., Gasull, A., Jarque, X.: Local and global phase portrait of equation z˙ = f (z). Discrete Contin. Dyn. Syst. 17(2), 309–329...
    • 23. Goldberg, L.R.: Catalan numbers and branched coverings by the Riemann sphere. Adv. Math. 85(2), 129–144 (1991)
    • 24. Graham, K.: On multiple Schramm-Loewner evolutions. J. Stat. Mech Theory Exp. 3(P03008), 21 (2007)
    • 25. Jahangoshahi, M., Lawler, G.F.: On the smoothness of the partition function for multiple SchrammLoewner evolutions. J. Stat. Phys. 173(5),...
    • 26. Kozdron, M. J., Lawler, G. F.: The configurational measure on mutually avoiding SLE paths, Universality and renormalization, Fields Inst....
    • 27. Kang, N. G., Makarov, N.: Gaussian free field and conformal field theory, Astérisque 353 (2013), viii+136
    • 28. Kang, N.-G., Makarov, N. G.: Calculus of conformal fields on a compact Riemann surface, arXiv:1708.07361 [math-ph] (2017)
    • 29. Kang, N.-G., Makarov, N. G.: Conformal field theory on the Riemann sphere and its boundary version for SLE, arXiv:2111.10057 [math-ph]...
    • 30. Kytölä, K., Peltola, E.: Pure partition functions of multiple SLEs. Comm. Math. Phys. 346(1), 237–292 (2016)
    • 31. Klimeš, M., Rousseau, C.: Remarks on rational vector fields on CP1. J. Dyn. Control Syst. 27(2), 293–320 (2021)
    • 32. Lawler, G. F.: Conformally invariant processes in the plane, School and Conference on Probability Theory, ICTP Lect. Notes, XVII, Abdus...
    • 33. Lawler, G. F.: Conformally invariant processes in the plane, Mathematical Surveys and Monographs, vol. 114. American Mathematical Society,...
    • 34. Lawler, G.F.: Partition functions, loop measure, and versions of SLE. J. Stat Phys. 134(5–6), 813–837 (2009)
    • 35. Lawler, G. F.: Schramm-Loewner evolution (SLE), Statistical mechanics, IAS/Park City Math. Ser., vol. 16, Amer. Math. Soc., Providence,...
    • 36. Lax, P.D.: Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Math. 21, 467–490 (1968)
    • 37. Lawler, G., Schramm, O., Werner, W.: Conformal restriction: the chordal case. J. Amer. Math. Soc. 16(4), 917–955 (2003)
    • 38. Muciño-Raymundo, J., Valero-Valdés, C.: Bifurcations of meromorphic vector fields on the Riemann sphere. Ergodic Theory Dynam. Syst. 15(6),...
    • 39. Moser, J.: Three integrable Hamiltonian systems connected with isospectral deformations. Adv. Math. 16, 197–220 (1975)
    • 40. Marshall, D.E., Rohde, S.: The Loewner differential equation and slit mappings. J. Amer. Math. Soc. 18(4), 763–778 (2005)
    • 41. Marshall, D., Rohde, S.,Wang, Y.: Piecewise geodesic jordan curves I: weldings, explicit computations, and schwarzian derivatives, arXiv:2202.01967...
    • 42. Miller, J., Sheffield, S.: Imaginary geometry I: interacting SLEs. Probab. Theory Relat. Fields 164(3–4), 553–705 (2016)
    • 43. Miller, J., Sheffield, S.: Imaginary geometry II: reversibility of SLEκ (ρ1; ρ2) for κ ∈ (0, 4). Ann. Probab. 44(3), 1647–1722 (2016)
    • 44. Miller, J., Sheffield, S.: Imaginary geometry III: reversibility of SLEκ for κ ∈ (4, 8). Ann Math (2). 184(2), 455–486 (2016)
    • 45. Miller, J., Sheffield, S.: Imaginary geometry IV: interior rays, whole-plane reversibility, and spacefilling trees. Probab. Theory Relat....
    • 46. Mukhin, E., Tarasov, V., Varchenko, A.: The B and M Shapiro conjecture in real algebraic geometry and the Bethe ansatz. Ann. Math. 170(2),...
    • 47. Polychronakos, A.P.: Feynman’s proof of the commutativity of the Calogero integrals of motion. Ann. Phys. 403, 145–151 (2019)
    • 48. Peltola, E., Hao, W.: Global and local multiple SLEs for κ ≤ 4 and connection probabilities for level lines of GFF. Comm. Math. Phys....
    • 49. Peltola, E., Wang, Y.: Large deviations of multichordal SLE0+, real rational functions, and zetaregularized determinants of Laplacians,...
    • 50. Ruijsenaars, S.N.M.: Action-angle maps and scattering theory for some finite-dimensional integrable systems: I: the pure soliton case....
    • 51. Schramm, O., Sheffield, S.: Contour lines of the two-dimensional discrete Gaussian free field. Acta Math. 202(1), 21–137 (2009)
    • 52. Schramm, O., Sheffield, S.: A contour line of the continuum Gaussian free field. Probab. Theory Relat. Fields 157(1–2), 47–80 (2013)
    • 53. Schramm, O., Wilson, D.B.: SLE coordinate changes. New York J. Math. 11, 659–669 (2005)
    • 54. Viklund, F., Wang, Y.: Interplay between Loewner and Dirichlet energies via conformal welding and flow-lines. Geom. Funct. Anal. 30(1),...
    • 55. Wang, Y.: The energy of a deterministic Loewner chain: reversibility and interpretation via SLE0+. J. Eur. Math. Soc. (JEMS) 21(7),...
    • 56. Wang, Y.: Equivalent descriptions of the Loewner energy. Invent. Math. 218(2), 573–621 (2019)
    • 57. Wilson, G.: Collisions of Calogero-Moser particles and an adelic Grassmannian, Invent. Math. 133(1), 1–41, With an appendix by I. G. Macdonald...
    • 58. Wu, H.: Hypergeometric SLE: conformal Markov characterization and applications. Commun. Math. Phys. 374(2), 433–484 (2020)
    • 59. Zhan, D.: Existence and uniqueness of nonsimple multiple SLE, arXiv:2308.13886 [math-PR] (2023)
Fundación Dialnet

Mi Documat

Opciones de artículo

Opciones de compartir

Opciones de entorno