The dimension of the SLE curves

Vincent Beffara (1)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

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:

Domaines: Probabilités [math.PR]

Vincent Beffara. The dimension of the SLE curves. Annals of Probability, 2008, 36 (4), pp.1421 - 1452. ⟨10.1214/07-AOP364⟩. ⟨hal-01693157⟩