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Fig. 6. The trace is about to hit the axis at x₀ and enclose a region. At the time this happens, the whole region including the point x₀ is mapped by g_t to the same point a_t.

The opposite is true for κ > 4: points on the real axis are continually colliding with the image a_t of the tip. This means that the trace is continually touching both itself and the real axis, at the same time engulfing whole regions. Moreover, since it is self-similar object, it does this on all scales, an infinite number of times within any finite neighbourhood! Eventually, the trace swallows the whole half plane: every point is ultimately mapped into a_t. For κ < 4 only the points on the trace itself suffer this fate. The case κ = 4 is more tricky: in fact the trace is then also a simple curve.

When κ is just above 4, the images of points on the real axis under g_t collide with a_t only when there happen to be rare events when the random force is strong enough to overcome the repulsion. When this happens, whole segments of the real axis are swallowed at one time, corresponding to the picture described above. Conversely, for large κ, the repulsive force is negligible except for very small x_t. In that case, two different starting points move with synchronised Brownian motions until the one which started off closer to the origin is swallowed. Thus, the real line is eaten up in a continuous fashion rather than piecemeal. There are no finite regions swallowed by the trace which are not on the trace itself. This means that the trace is space-filling: γ intersects every neighbourhood of every point in the upper half plane. We shall argue later (Section 4.3.1) that the fractal dimension of the trace is d_f = 1 + κ/8 for κ 8 and 2 for κ 8. Thus, it becomes space-filling for all κ 8.

3.4.2. SLE duality

For κ > 4 the curve continually touches itself and therefore its hull K_t contains earlier portions of the trace (see Fig. 4). However, the frontier of K_t (i.e., the boundary of H K_t, minus any portions of the real axis), is by definition a simple curve. A beautiful result, first suggested by Duplantier [20], and proved by Beffara [21] for the case κ = 6, is that locally this curve is an , with

For example, the exterior of a percolation cluster contains many ‘fjords’ which, on the lattice, are connected to the main ocean by a neck of water which is only a finite number of lattice spacings wide. These are sufficiently frequent and the fjords macroscopically large that they survive in the continuum limit. SLE₆ describes the boundaries of the clusters, including the coastline of all the fjords. However, the coastline as seen from the ocean is a simple curve, which is locally SLE_8/3, the same as that conjectured for a self-avoiding walk. This suggests, for example, that locally the frontier of a percolation cluster and a self-avoiding walk are the same in the scaling limit. In Section 5, we show that SLE_κ and correspond to CFTs with the same value of the central charge c.

3.5. Special values of κ

3.5.1. Locality

(This subsection and the next are more technical and may be omitted at a first reading). We have defined SLE in terms of curves which connect the origin and infinity in the upper half plane. Property 3.2 then allows us to define it for any pair of boundary points in any simply connected domain, by a conformal mapping. It is interesting to study how the variation of the domain affects the SLE equation. Let A be a simply connected region connected to the real axis which is at some finite distance from the origin (see Fig. 7). Consider a trace γ_t, with hull K_t, which grows from the origin according to SLE in the domain H A. According to Property 3.2, we can do this by first making a conformal mapping h₀ which removes A, and then a map which removes the image . This would be described by SLE in h₀ (H A), except that the Loewner ‘time’ would not in general be the same as t. However, another way to think about this is to first use a SLE map g_t in H to remove K_t, then another map, call it h_t, to remove g_t (A). Since both these procedures end up removing K_t A, and all the maps are assumed to be normalised at infinity in the standard way (15), they must be identical, that is (see Fig. 7). If g_t maps the growing tip τ_t to a_t, then after both mappings it goes to ã_t = h_t (a_t). We would like to understand the law of ã_t.

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Fig. 7. An SLE hull in H A and two different ways of removing it: either by first removing A through h₀ and then using a Loewner map in the image of H A; or by removing K_t first with g_t and then removing the image of A with h_t. Since all maps are normalised, this diagram commutes.

Rather than working this out in full generality (see for example [12]), let us suppose that A is a short vertical segment (x, x + i ) with x, and that t = dt is infinitesimal. Then, under g_dt, x → x + 2dt/x and → (1 − 2dt/x²). The map that removes this is (see (16))

To find ã_dt, we need to set in this expression. Carefully expanding this to first order in dt, remembering that (dB_t)² = dt, and also taking the first non-zero contribution in /x, gives after a few lines of algebra

The factor in front of the stochastic term may be removed by rescaling dt: this restores the correct Loewner time. But there is also a drift term, corresponding to the effect of A. For κ < 6 we see that the SLE is initially repelled from A. From the point of view of the exploration process for the Ising model discussed in Section 2.1.1, this makes sense: if the spins along the positive real axis and on A are fixed to be up, then the spin just above the origin is more likely to be up than down, and so γ is more likely to turn to the left.

For κ = 6, however, this is no longer the case: the presence of A does not affect the initial behaviour of the curve. This is a particular case of the property of locality when κ = 6, which states that, for any A as defined above, the law of K_t in H A is, up to a time reparametrisation, the same as the law of K_t in H, as long as K_t ∩ A = . That is, up to the time that the curve hits A, it does not know its there. Such a property would be expected for the cluster boundaries of uncorrelated Ising spins on the lattice, i.e., percolation. This is then consistent with the identification of percolation cluster boundaries with SLE₆.

3.5.2. Restriction

It is also interesting to work out how the local scale transforms in going from a_t to ã_t. A measure of this is . A similar calculation starting from (22) gives, in the same limit as above

Now something special happens when κ = 8/3. The drift term in d (h′ (a_t)) does not then vanish, but if we take the appropriate power it does. This implies that the mean of is conserved. Now at t = 0 it takes the value , where Φ_A = h₀ is the map that removes A. If K_t hits A at time T it can be seen from (22) that . On the other hand, if it never hits A then . Therefore, gives the probability that the curve γ does not intersect A.

This is a remarkable result in that it depends only on the value of at the starting point of the SLE (assuming of course that Φ_A is correctly normalised at infinity). However, it has the following even more interesting consequence. Let . Consider the ensemble of all SLE_8/3 in H, and the sub-ensemble consisting of all those curves γ which do not hit A. Then the measure on the image in H is again given by SLE_8/3. The way to show this is to realise that the measure on γ is characterised by the probability P (γ ∩ A′ = ) that γ does not hit A′ for all possible A′. The probability that does not hit A′, given that γ does not hit A, is the ratio of the probabilities and P (γ ∩ A = ). By the above result, the first factor is the derivative at the origin of the map which removes A then A′, while the second is the derivative of the map which removes A. Thus

Since this is true for all A′, it follows that the law of given that γ does not intersect A is the same as that of γ. This is called the restriction property. Note that while, according to Property 3.2, the law of an SLE in any simply connected subset of H is determined by the conformal mapping of this subset to H, the restriction property is stronger than this, and it holds only when κ = 8/3.

We expect such a property to hold for the continuum limit of self-avoiding walks, assuming it exists. On the lattice, every walk of the same length is counted with the same weight. That is, the measure is uniform. If we consider the sub-ensemble of such walks which avoid a region A, the measure on the remainder should still be uniform. This will be true if the restriction property holds. This supports the identification of self-avoiding walks with SLE_8/3.

3.6. Radial SLE and the winding angle

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.

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