3.6.1. Identification with lattice models

This result allows use to make a tentative identification with the various realisations of the O (n) model described in Section 2.2. We have, using (27), n = −2 cos(4π/κ) with 2  κ  4 describing the critical point at x_c, and 4 < κ  8 corresponding to the dense phase. Some important special cases are therefore:

• κ = −2: loop-erased random walks (proven in [24]);

• κ = 8/3: self-avoiding walks, as already suggested by the restriction property, Section 3.5.2; unproven, but see [22] for many consequences;

• κ = 3: cluster boundaries in the Ising model, however, as yet unproven;

• κ = 4: BCSOS model of roughening transition (equivalent to the 4-state Potts model and the double dimer model), as yet unproven; also certain level lines of a gaussian random field and the ‘harmonic explorer’ (proven in [23]); also believed to be dual to the Kosterlitz–Thouless transition in the XY model;

• κ = 6: cluster boundaries in percolation (proven in [7]);

• κ = 8: dense phase of self-avoiding walks; boundaries of uniform spanning trees (proven in [24]).

It should be noted that no lattice candidates for κ > 8, or for the dual values κ < 2, have been proposed. This possibly has to do with the fact that, for κ > 8, the SLE trace is not reversible: the law on curves from r₁ to r₂ is not the same as the law obtained by interchanging the points. Evidently, curves in equilibrium lattice models should satisfy reversibility.

4. 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.

4.1. Schramm’s formula

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.