Radial SLE and the winding angle

Date: 2015-10-07; view: 440.

So far we have discussed a version of SLE that gives a conformally invariant measure on curves which connect two distinct boundary points of a simply connected domain . For this reason it is called chordal SLE. There are variants which describe other situations. For example, one could consider curves γ which connect a given point r₁ on the boundary to an interior point r₂. The Riemann mapping theorem allows us to map conformally onto another simple connected domain, with r₂ being mapped to any preassigned interior point. It is simplest to choose for the standard domain the unit disc U, with r₂ being mapped to the origin. So we are considering curves γ which connect a given point e^iθ₀ on the boundary with the origin. As before, we may consider growing the curve dynamically. Let K_t be the hull of that portion which exists up to time t. Then there exists a conformal mapping g_t which takes U K_t to U, such that g_t (0) = 0. There is one more free parameter, which corresponds to a global rotation: we use this to impose the condition that is real and positive. One can then show that, as the curve grows, this quantity is monotonically increasing, and we can use this to reparametrise time so that . This normalised mapping then takes the growing tip τ_t to a point e^iθ_t on the boundary.

Loewner's theorem tells us that , when expressed as a function of g_t (z), should be holomorphic in apart from a simple pole at e^iθ_t. Since g_t preserves the unit circle outside should be pure imaginary when |g_t (z)| = 1, and in order that , it should approach 1 as g_t (z) → 0. The only possibility is

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This is the radial Loewner equation. In fact this is the version considered by Löewner [4].

It can now be argued, as before, that given lk and lk (suitably reworded to cover the case when r₂ is an interior point) together with reflection, θ_t must be proportional to a standard Brownian motion. This defines radial SLE. It is not immediately obvious how the radial and chordal versions are related. However, it can be shown that, if the trace of radial SLE hits the boundary of the unit disc at e^iθ_t1 at time t₁, then the law of K_t in radial SLE, for t < t₁, is the same chordal SLE conditioned to begin at e^iθ(0) and end at e^iθ_t1, up to a reparametrisation of time. This assures us that, in using the chordal and radial versions with the same κ, we are describing the same physical problem.

However, one feature that the trace of radial SLE possesses which chordal SLE does not is the property that it can wind around the origin. The winding angle at time t is simply θ_t − θ₀. Therefore, it is normally distributed with variance κt. At this point we can make a connection to the Coulomb gas analysis of the O (n) model in Section 2.4.1, where it was shown that the variance in the winding angle on a cylinder of length L is asymptotically (4/g)L. A semi-infinite cylinder, parametrised by w, is conformally equivalent to the unit disc by the mapping z = e^−w. Asymptotically, Re w → Re w − t under Loewner evolution. Thus, we can identify L t and hence