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|All Authors / Contributors:||
Julien Dubédat; Wendelin Werner; Université de Paris-Sud.
|Notes:||Thèse rédigée majoritairement en anglais.|
|Description:||118 p. : ill. ; 30 cm.|
|Responsibility:||Julien Dubédat ; sous la dir. de Wendelin Werner.|
The subject of this work is the study of several properties of the Schramm-Loewner Evolution (SLE), particularly in relation with the scaling limit of critical plane percolation. Firstly, one considers holomorphic martingales of SLE, the associated plane geometries, and, in the case of SLE(6), the family of holonomic systems satisfied by some natural events for critical percolation. Then, one studies a family of plane reflected Brownian motions connected with SLE; in particular, one gives an interpretation of Watts formula, which was enunciated in the critical percolation context, in terms of reflected brownian motion. One also examinates percolation formulae attached to annular configurations; the convergence of critical percolation interfaces to SLE(6) leads to an expression of these questions as first exit problems for a Markov process taking values in a modular space. The duality conjecture for SLE pertains to the law of the SLE boundary when this SLE is not a simple path; one formulates precise conjectures based on the analysis of restriction formulae. Lastly, one proves the existence of excursion decompositions for SLE with respect to frontier points and cutpoints respectively. Incidentally, this leads to a rigorous proof of a percolation formula, Watts formula.