The dimension of the SLE curves

By Vincent Beffara. 2002

273 citations recorded by the source corpus.

Let γ be the curve generating a Schramm–Loewner Evolution (SLE) process, with parameter κ ≥ 0. We prove that, with probability one, the Haus-dorff dimension of γ is equal to Min(2, 1 + κ/8). Introduction. It has been conjectured by theoretical physicists that various lattice models in statistical physics (such as percolation, Potts model, Ising model, uniform spanning trees), taken at their critical point, have a continuous confor-mally invariant scaling limit when the mesh of the lattice tends to 0. Recently, Oded Schramm [15] introduced a family of random processes which he called Stochastic Loewner Evolutions (or SLE), that are the only possible conformally invariant scaling limits of random cluster interfaces (which are very closely related to all above-mentioned models). An SLE process is defined using the usual Loewner equation, where the driving function is a time-changed Brownian motion. More specifically, in the present paper we will be mainly concerned with SLE in the upper-half plane (sometimes called chordal SLE), defined by the following PDE:

Subfields: Condensed Matter Physics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1f6886f6655546c8e5a37fef00697a16c1ddae19

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Citations in the corpus

Towards conformal invariance of 2D lattice models (2007, background): “Moreover, Vincent Beffara [7] has proved that the Hausdorff dimension of the SLE (κ) curve is almost surely min (1 + κ/8, 2).”
Conformal loop ensembles: the Markovian characterization and the loop-soup construction (2010, background): “lim c→c0− P (d(c) ≥ x) ≥ lim κ→κ0− P (d(γ, [1, 2]) ≥ x) ≥ P 0(d(γ, [1, 2]) ≥ 2x).”
Almost sure multifractal spectrum for the tip of an SLE curve (2009, methodology): “It was proved by Beffara [1] that d is the Hausdorff dimension of the path γ[0, 2].”
SLE curves and natural parametrization (2010, methodology): “ond moment estimates for the reverse Loewner ﬂow. In this paper, we prove that the deﬁnition in [11] is valid for κ<8. Instead of using the reverse Loewner ﬂow, we use a diﬃcult estimate of Beﬀara [1] on the forward Loewner ﬂow to establish the necessary uniform integrability to apply the Doob–Me”
Imaginary geometry IV: interior rays, whole-plane reversibility, and space-filling trees (2013, background): “The value of d(θ1) = d(0) is the dimension of ordinary chordal SLEκ: 1 + κ 8 [1].”
Duality of chordal SLE (2007, background): “For κ ≤ 8, the Hausdorff dimension of an SLE(κ) trace was proved to be 1 + κ/8 [3].”