Hausdorff Dimension and mean porosity

SLE κ is a random growth process based on Loewner's equation with driving parameter a one-dimensional Brownian motion running with speed κ. This process is intimately connected with scaling limits of percolation clusters and with the outer boundary of Brownian motion, and is conjectured to correspond to scaling limits of several other discrete processes in two dimensions.The present paper attempts a first systematic study of SLE. It is proved that for all κ = 8 the SLE trace is a path; for κ ∈ [0, 4] it is a simple path; for κ ∈ (4, 8) it is a self-intersecting path; and for κ > 8 it is space-filling.It is also shown that the Hausdorff dimension of the SLE κ trace is almost surely (a.s.) at most 1 + κ/8 and that the expected number of disks of size ε needed to cover it inside a bounded set is at least ε −(1+κ/8)+o(1) for κ ∈ [0, 8) along some sequence ε 0. Similarly, for κ ≥ 4, the Hausdorff dimension of the outer boundary of the SLE κ hull is a.s. at most 1 + 2/κ, and the expected number of disks of radius ε needed to cover it is at least ε −(1+2/κ)+o(1) for a sequence ε 0.

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“…In fact, this follows from [KR97], but we include a short proof here, for the convenience of the reader.…”

Section: Proof Of Theoremmentioning

confidence: 98%

Basic properties of SLE

Rohde¹,

Schramm²

2005

Ann. Math.

Self Cite

375

501

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“…The value of t 0 depends on the growth of the number of Whitney cubes of Ω of sidelength at least 2 −k , and so is controlled by d ≡ dim M ∂Ω, the Minkowski dimension of ∂Ω (see [27], [28], [8]), which easily leads to the inequality t 0 ≤ d − n (however, t 0 may equal −1 even if d = n). Since the boundary of a QHBC domain Ω has Minkowski dimension strictly less than n (see [30], [25]), it follows that t 0 ∈ [−1, 0) as claimed. The rest of the theorem, except for the sharpness of ϕ s , follows from Theorems 3.4 and 3.5 and the comments after those theorems.…”

Section: Proof Ofmentioning

confidence: 90%

Weighted Trudinger-type inequalities

Buckley¹,

O'Shea²

1999

Indiana Univ. Math. J.

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Abstract. We establish sharp inequalities of Trudinger-type (with non-standard Young functions) on general domains Ω with respect to measures ν, µ. Our results are new even when Ω is a Euclidean ball, and ν, µ are defined in terms of powers of distance to the boundary. In the Euclidean case, sharpness results are proved for the Young functions involved and the class of domains considered. New characterizations of QHBC domains are given.

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“…The proof of Theorem 1.1 is based on a variant of mean porosity arguments developed in [10], [13] and [8]. The example showing the sharpness is a modification of the example given in [9].…”

Section: Theorem 12mentioning

confidence: 99%

“…Our proof is based on a covering argument similar to the ones in [10], [13] and [8]. In the calculations below we need to use (9) and (10) roughly speaking only for the tail ends of the approximating sequences.…”

Section: Remarkmentioning

confidence: 99%