Introduction to Schramm–Loewner evolutions

we mean an equivalence class of paths modulo reparametrizations. Write CD for the set of chords in D. Given a path γ, we write [γ] for the associated chord; more generally, if γ is a continuous map from any compact subinterval I of [0,∞] to Û , starting at z0 and ending at z1, we can obtain a chord [γ] in D by choosing an increasing homeomorphism φ : [0, 1] → I and setting [γ] = [γ ◦ φ]. Write D for the set of triples D = (U, z0, z1), where U is a simply connected proper (that is, not ∅ or C) planar domain and where z0, z1 are distinct points in Û \ U . For D ∈ D, by a filling in D we mean a closed, connected, simply connected subset of Û containing z0 and z1. Write SD for the set of fillings in D. We shall study some families of probability measures (μD : D ∈ D) where, for each D, μD is the distribution of a random chord or a random filling in D. We defer until Sections 7 and 9 the specification of σ-algebras on CD and SD needed to make these notions precise. For D,D′ ∈ D, a conformal isomorphism Φ : D → D′ is a conformal isomorphism U → U ′ with Φ(z0) = z′ 0 and Φ(z1) = z′ 1. The set of conformal automorphisms of D forms a one-parameter group, whose elements we call scalings, by analogy with the case of the upper half-plane H with boundary points 0 and ∞, where the conformal automorphisms are given by Φλ(z) = λz for λ ∈ (0,∞). We say that the measure μD is scale-invariant if, whenever X ∼ μD, then also Φ(X) ∼ μD, for any scaling Φ.