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Christophe Raymond Garban; Wendelin Werner; Université de Paris-Sud. Faculté des Sciences d'Orsay (Essonne).
|Anmerkungen:||Thèse rédigée entièrement en anglais.
Résumé étendu en français (p. 1-44).
|Beschreibung:||1 vol. (VI-266 p.) : ill. ; 30 cm.|
|Verfasserangabe:||Christophe Garban ; [sous la direction de] Wendelin Werner.|
This thesis focuses on some properties of critical planar percolation as well as SLE Processes. The first chapter deals with the expected area of a planar Brownian Loop of time one. This expected area is computed using SLE techniques and happens to be Pi over five. The second chapter presents an Analog of Makarov Theorem about SLE curves and leads to the almost sure continuity of the SLE curves in arbitrary simply connected domains. The Third chapter (which, in some sense is the main one) deals with properties of dynamical peroclation. It is proved for instance that the set of exceptional times (where an inifnite cluster appears) has fractal dimension 31/36 on the triangular lattice (at p_c =1/2). The existence of these exceptional times is also proved in the case of the square grid Z^2. These questions are related to the so called phenomenon of "noise sensitivity" of percolation. Various sharp results are provided about this noise sensitivity in a general setting. The proofs rely on Discrete Fourier analysis. The Last chapter is about the scaling limit of "near-critical" percolation. We present some results which will be the key steps in a forthcoming proof of the existence and uniqueness of this scaling limit. These results apply as well to the setting of the scaling limit of dynamical percolation.