Schramm's formula

Date: 2015-10-07; view: 436.

Calculating with SLE

SLE shows that the measure on the continuum limit of single curves in various lattice models is given in terms of one-dimensional Brownian motion. However, it is not at all clear how thereby to deduce interesting physical consequences. We first describe two relatively simple computations in two-dimensional percolation which can be done using SLE.

Given a curve γ connecting two points r₁ and r₂ on the boundary of a domain , what is the probability that it passes to the left of a given interior point? This is not a question which is natural in conventional approaches to critical behaviour, but which is very simply answered within SLE [25].

As usual, we can consider to be the upper half plane, and take r₁ = a₀ and r₂ to be at infinity. The curve is then described by chordal SLE starting at a₀. Label the interior point by the complex number ζ.

Denote the probability that γ passes to the left of ζ by (we include the dependence on to emphasise the fact that this is a not a holomorphic function). Consider evolving the SLE for an infinitesimal time dt. The function g_dt will map the remainder of γ into its image γ′, which, however, by lk and lk, will have the same measure as SLE started from . At the same time, ζ → g_dt (ζ) = ζ + 2dt/(ζ − a₀). Moreover, γ′ lies to the left of ζ′ iff γ lies to the left of ζ. Therefore

(28)

where the average … is over all realisations of Brownian motion dB_t up to time dt. Taylor expanding, using dB_t = 0 and (dB_t)² = dt, and equating the coefficient of dt to zero gives

(29)

Thus, P satisfies a linear second-order partial differential equation, typical of conditional probabilities in stochastic differential equations.

By scale invariance P in fact depends only on the angle θ subtended between ζ − a₀ and the real axis. Thus, (29) reduces to an ordinary second-order linear differential equation, which is in fact hypergeometric. The boundary conditions are that P = 0 and 1 when θ = π and 0, respectively, which gives (specialising to κ = 6)

(30)

Note that this may also be written in terms of a single quadrature since one solution of (29) is P = const.