Hausdorff dimensions for SLE6

By Vincent Beffara. 2002

93 citations recorded by the source corpus.

We prove that the Hausdorff dimension of the trace of SLE6 is almost surely 7/4 and give a more direct derivation of the result (due to Lawler–Schramm–Werner) that the dimension of its boundary is 4/3. We also prove that, for all κ<8, the SLEκ trace has cut-points.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0f8f966c89a3fc5c033668e7fdfbd1fda2601712

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Percolation Theory (2005, background): “Percolation theory Vincent Beffara, Vladas Sidoravicius To cite this version: Vincent Beffara, Vladas Sidoravicius.”
SLE and the free field: partition functions and couplings (2007, background): “Recall in particular that for κ ∈ (4, 8), SLE develops cutpoints ([3, 8]).”
Random planar curves and Schramm-Loewner evolutions (2003, background): “Various exponents are derived in for instance in [86, 87, 88, 92, 118, 14 , 15].”
Duality of chordal SLE (2007, background): “From Theorem 5 in [ 3 ], if κ > 8 and T > 0 is a constant, then a.s.”
The dimension of the SLE curves (2002, methodology): “The main theorem then follows from the zero-one law derived in [ 2 ], namely:”
Critical percolation exploration path and SLE6: a proof of convergence (2006, background): “The significance of the notion of admissible is that certain domains arising naturally in the proof of Theorem 4 are not Jordan but are admissible; this is because the hulls Kt generated by chordal SLE6 paths have cut-points [4]—see Fig.”
Excursion decompositions for SLE and Watts' crossing formula (2004, background): “Hausdorff dimension of frontier times has been derived in [1].”
Recent advances in percolation theory and its applications (2015, background): “In the dense phase with κ > 4, there is a duality conjecture stating that the exterior frontier of an SLEκ hull looks locally as SLEκ̃ with κ̃ = 16/κ [135, 260].”
SLE for theoretical physicists (2005, background): “A beautiful result, first suggested by Duplantier [20], and proved by Beffara [21] for the case j = 6, is that locally this curve is an SLE~j, with ~j ¼ 16=j. ð21Þ For example, the exterior of a percolation cluster contains many fjords which, on the lattice, are connected to the main ocean by a neck”