Schramm–Loewner evolution

Date: 2015-10-07; view: 351.

In the case that we are interested in, γ is a random curve, so that a_t is a random continuous function. What is the measure on a_t? This is answered by the following remarkable result, due to Schramm [5]:

Theorem 3.1

If Properties 3.1–3.2 hold, together with reflection symmetry, then a_t is proportional to a standard Brownian motion.

That is

(19)

so that a_t = 0, (a_t1-a_t2)² =κ|t₁-t₂|. The only undetermined parameter is κ, the diffusion constant. It will turn out that different values of κ correspond to different universality classes of critical behaviour.

The idea behind the proof is once again simple. As before, consider growing the curve for a time t₁, giving γ₁, and denote the remainder γ γ₁ = γ₂. Property 3.1 tells us that the conditional measure on γ₂ given γ₁ is the same as the measure on γ₂ in the domain H K_t1, which, by Property 3.2, induces the same measure on g_t1(γ₂) in the domain H, shifted by a_t1. In terms of the function a_t this means that the probability law of a_t-a_t1, for t > t₁, is the same as the law of a_t-t1. This implies that all the increments Δ_n ≡ a_(n+1)δt − a_nδt are independent identically distributed random variables, for all δt > 0. The only process that satisfies this is Brownian motion with a possible drift term: . Reflection symmetry then implies that α = 0.