Brownian beads
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In this article, it was shown that the past and future of halfplane Brownian motion at certain cutpoints are independent of each other after a conformal transformation, and that the pieces between cutpoints form a Poisson process with respect to a local time.Abstract:
We show that the past and future of half-plane Brownian motion at certain cutpoints are independent of each other after a conformal transformation Like in Ito's excursion theory, the pieces between cutpoints form a Poisson process with respect to a local time The size of the path as a function of this local time is a stable subordinator whose index is given by the exponent of the probability that a stretch of the path has no cutpoint The index is computed and equals 1/2read more
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Journal ArticleDOI
Conformal restriction: The chordal case
TL;DR: In this article, the authors characterize and describe all random subsets K of a given simply connected planar domain (the upper halfplane Η, say) which satisfy the conformal restriction property, i.e., K connects two fixed boundary points (0 and ∞, say).
Journal ArticleDOI
The Brownian loop soup
TL;DR: In this article, a conformally invariant measure on unrooted Brownian loops in the plane is defined and some properties of its properties are studied. But this measure is restricted to loops rooted at a boundary point of a domain.
Posted Content
Random planar curves and Schramm-Loewner evolutions
TL;DR: In this paper, the Schramm-Loewner evolution (SLE) is used to define a set of random curves, including uniform spanning trees, self-avoiding walks, critical percolation, and loop-erased random walks.
Book ChapterDOI
Wendelin Werner: Random Planar Curves and Schramm-Loewner Evolutions
TL;DR: The critical percolation exploration process of the half-plane half-plane is described in this article. But the authors do not specify a method to find the critical locations of critical intersections.
Journal ArticleDOI
SLE(κ,ρ) martingales and duality
TL;DR: In particular, this article derived certain restriction properties that lead to a strong duality conjecture, which is an identity in law between the outer boundary of a variant of the SLE(κ) process for κ≥4 and a variation of the Schramm-Loewner process (SLE(16/κ).