Mouvement brownien plan, SLE, invariance conforme et dimensions fractales

Beffara, Vincent

Directeur de thèse : Werner, Wendelin

Membres du jury : Le Jan, Yves ; Duplantier, Bertrand ; Kenyon, Richard ; Lawler, Gregory F. ; Le Gall, Jean-François ; Werner, Wendelin ; Grimmett, Geoffrey R.

Thèses d'Orsay, no. 634 (2003) , 138 p.

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@phdthesis{BJHTUP11_2003__0634__A1_0,
     author = {Beffara, Vincent},
     title = {Mouvement brownien plan, {SLE,} invariance conforme et dimensions fractales},
     series = {Th\`eses d'Orsay},
     publisher = {Universit\'e Paris XI UFR scientifique d'Orsay},
     number = {634},
     year = {2003},
     language = {fr},
     url = {http://archive.numdam.org/item/BJHTUP11_2003__0634__A1_0/}
}

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Beffara, Vincent. Mouvement brownien plan, SLE, invariance conforme et dimensions fractales. Thèses d'Orsay, no. 634 (2003), 138 p. http://numdam.org/item/BJHTUP11_2003__0634__A1_0/

Bibliographie
Cité par

[1] L. V. Ahlfors, Conformal Invariants, Topics in Geometric Function Theory, McGraw- Hill, New York, 1973. | MR | Zbl

[2] R. F. Bass, Diffusions and elliptic operators. Probability and Its Applications, Springer, 1998. | MR | Zbl

[3] R. F. Bass and K. Burdzy, Cutting Brownian Paths, vol. 657 of Memoirs of the American Mathematical Society, AMS, 1999. | Zbl | MR

[4] R. F. Bass and K. Burdzy, The supremum of Brownian local times on Holder curves, Annales de l'institut Henri Poincarré, 37 (2001), pp. 627-642. | MR | Zbl | Numdam | DOI

[5] V. Beffara, On conformally invariant subsets of the planar Brownian curve, Preprint, arXiv :math.PR/0105192, (2001). | MR | Zbl | Numdam

[6] V. Beffara, $S L E_{κ}$ when $κ \to 0 or κ \to \infty$ , In preparation, (2002).

[7] K. Burdzy, Cut points on Brownian paths, Ann. Probab., 17 (1989), pp. 1012-1036. | MR | Zbl | DOI

[8] L. Carleson, On the support of harmonic measure for sets of Cantor type, Ann. Acad. Sci. Fenn. Ser. A I Math, 10 (1985), pp. 113-123. | MR | Zbl | DOI

[9] L. Carleson and P. W. Jones, On coefficient problems for univalent functions and conformal dimension, Duke Mathematical Journal, 66 (1992), pp. 169-206. | MR | Zbl | DOI

[10] L. Carleson, P. W. Jones, and J.-C. Yoccoz, Julia and John, Bol. Soc. Bras. Mat., 25 (1994), pp. 1-30. | MR | Zbl | DOI

[11] M. C. Cranston and T. S. Mountford, An extension of a result of Burdzy and Lawler, Probab. Theory Relat. Fields, 89 (1991), pp. 487-502. | MR | Zbl | DOI

[12] A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Springer, 2 ed., 1998. | MR | Zbl

[13] B. Diu, C. Guthmann, D. Lederer, and B. Roulet, Éléments de Physique Statistique, Hermann, 1989.

[14] B. Duplantier, Conformally invariant fractals and potential theory, Phys. Rev. Lett., 84 (2000), pp. 1363-1367. | MR | Zbl | DOI

[15] P. L. Duren, Univalent Functions, Springer-Verlag, 1983. | MR | Zbl

[16] S. N. Evans, On the Hausdorff dimension of Brownian cone points, Math. Proc. Camb. Phil. Soc., 98 (1985), pp. 343-353. | MR | Zbl | DOI

[17] R. Friedrich and W. Werner, Conformal fields, restriction properties, degenerate representations and SLE, Preprint, arXiv :math. PR/0209382, (2002). | MR | Zbl | DOI

[18] K. Gawędzki, Lectures on conformal field theory. | Zbl

[19] G. Grimmett, Percolation, Springer-Verlag, 1989. | MR | Zbl | DOI

[20] P. W. Jones and N. G. Makarov, Density properties of harmonic measure. Annals of Mathematics, 142 (1995), pp. 427-455. | MR | Zbl

[21] J.-P. Kahane, Some Random Series of Functions, vol. 5 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2 ed., 1993. | Zbl

[22] R. Kaufman, Une propriété métrique du mouvement brownien, C.R. Acad. Sci., A 268 (1969), pp. 727-728. | MR | Zbl

[23] R. Langlands, Y. Pouillot, and Y. Saint-Aubin, Conformal invariance in two-dimensional percolation, Bulletin of the A.M.S., 30 (1994). | MR | Zbl | DOI

[24] G. F. Lawter, The dimension of the frontier of planar Brownian motion, Elect. Comm, in Probab., 1 (1996), pp. 29-47. | MR | Zbl

[25] G. F. Lawter, Hausdorff dimension of cut points for Brownian motion. Electronic Journal of Probability, 1 (1996), pp. 1-20. | MR | Zbl

[26] G. F. Lawter, Strict concavity of the intersection exponent for Brownian motion in two and three dimensions, Mathematical Physics Electronic Journal, 4 (1998). | MR | Zbl

[27] G. F. Lawter, Geometric and fractal properties of Brownian motion and random walk paths in two and three dimensions, in Random Walks (Budapest, 1998), vol. 9 of Bolyai Soc. Math. Stud., 1999, pp. 219-258. | MR | Zbl

[28] G. F. Lawler and E. E. Puckette, The intersection exponent for simple random walks, Comb. Prob. Comp., 9 (2000), pp. 441-464. | MR | Zbl | DOI

[29] G. F. Lawler, O. Schramm, and W. Werner, Conformal restriction : The chordal case, Preprint, arXiv :math.PR/0209343, (2001). | MR | Zbl

[30] G. F. Lawler, O. Schramm, and W. Werner, The dimension of the Brownian frontier is 4/3, Math. Rev. Lett., 8 (2001), pp. 13-24. | MR | Zbl | DOI

[31] G. F. Lawler, O. Schramm, and W. Werner, Values of Brownian intersection exponents I : Half-plane exponents, Acta Mathematica, 187 (2001), pp. 237-273. | MR | Zbl | DOI

[32] G. F. Lawler, O. Schramm, and W. Werner, Values of Brownian intersection exponents II : Plane exponents, Acta Mathematica, 187 (2001), pp. 275-308. | MR | Zbl | DOI

[33] G. F. Lawler, O. Schramm, and W. Werner, Analyticity of intersection exponents for planar Brownian motion, To appear in Acta Mathematica, 188 (2002). | MR | Zbl

[34] G. F. Lawler, O. Schramm, and W. Werner, Conformal invariance of planar loop-erased random walks and uniform spanning trees, Preprint, arXiv :math.PR/0112234, (2002). | MR | Zbl

[35] G. F. Lawler, O. Schramm, and W. Werner, On the scaling limit of planar self-avoiding walks, Preprint, arXiv :math.PR/0204277, (2002). | MR | Zbl

[36] G. F. Lawler, O. Schramm, and W. Werner, Values of Brownian intersection exponents III : Two-sided exponents, Annales Inst. Henri Poincaré PR, 38 (2002), pp. 109-123. | MR | Zbl | Numdam | DOI

[37] G. F. Lawler and W. Werner, Intersection exponents for planar Brownian motion, Ann. Probab., (1999), pp. 1601-1642. | MR | Zbl

[38] J.-F. Le Gall, Some properties of planar Brownian motion, in École d'été de Probabilités de Saint-Flour XX-1990, vol. 1527 of Lecture Notes in Mathematics, Springer, 1992. | MR | Zbl

[39] N. Madras and A. D. Sokal, The pivot algorithm : A highly efficient Monte-Carlo method for the self-avoiding walk, Journal of Statistical Physics, 50 (1988), pp. 109-186. | MR | Zbl | DOI

[40] P. Mattila, Geometry of sets and measures in Euclidean spaces, vol. 44 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, 1995. | MR | Zbl

[41] C. Pommerenke, Boundary Behaviour of Conformal Maps, Springer, 1992. | MR | Zbl | DOI

[42] S. Rohde and O. Schramm, Basic properties of SLE, Preprint, arXiv :math.PR/0106036, (2001). | Zbl

[43] O. Schramm, Scaling limits of loop-erased random walks and uniform spanning trees, Israel Journal of Mathematics, 118 (2000), pp. 221-288. | MR | Zbl | DOI

[44] S. Smirnov, Critical percolation in the plane : Conformal invariance, Cardy's formula, scaling limits, C. R. Acad. Sci. Paris Ser. I Math., 333 (2001), pp. 239-244. | MR | Zbl

[45] S. Smirnov and W. Werner, Critical exponents for two-dimensional percolation, Mathematical Research Letters, 8 (2001), pp. 729-744. | MR | Zbl | DOI

[46] W. Werner, On Brownian disconnection exponents, Bernoulli, 1 (1995), pp. 371-380. | MR | Zbl | DOI

[47] W. Werner, Bounds for disconnection exponents, Elect. Comm, in Probab., 1 (1996), pp. 19-28. | MR | Zbl

[48] W. Werner, Critical exponents, conformal invariance and planar Brownian motion, in Proceedings of the 3rd European Mathematical Congress, Birkhäuser, 2000. | MR | Zbl

[49] W. Q. Yan, On univalent functions in multiply connected domains, Journal of the Beijing Institute of Technology, 3 (1994), pp. 99-113. | MR | Zbl