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We have seen that, on the lattice, the curves γ may be ‘grown’ through a discrete exploration process. The Loewner process is the continuum version of this. Because of Property 3.2 it suffices to describe this in a standard domain , which is taken to be the upper half plane H, with the points r₁ and r₂ being the origin and infinity, respectively.

The first thing to notice is that, although on the honeycomb lattice the growing path does not intersect itself, in the continuum limit it might (although it still should not cross itself.) This means that there may be regions enclosed by the path which are not on the path but nevertheless are not reachable from infinity without crossing it. We call the union of the set of such points, together with the curve itself, up to time t, the hull K_t. (This is a slightly different usage of this term from that in the physics percolation literature.) It is the complement of the connected component of the half plane which includes ∞, itself denoted by H K_t. See Fig. 4.

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Fig. 4. Schematic view of a trace and its hull.

Another property which often holds in the half-plane is that of reflection invariance: the distribution of lattice paths starting from the origin and ending at ∞ is invariant under x → −x. For the lattice paths in the O (n) model discussed in Section 2.2 this follows from the symmetry of the underlying weights, but for the boundaries of the FK clusters in the Potts model it is a consequence of duality. Not all simple curves in lattice models have this property. For example, if we consider the three-state Potts model in which the spins on the negative and positive real axes are fixed to different values, there is a simple lattice curve which forms the outer boundary of the spin cluster containing the positive real axis. This is not the same as the boundary of the spin cluster containing the negative real axis, and it is not in general symmetric under reflections.

Since H K_t is simply connected, by the Riemann mapping theorem it can be mapped into the standard domain H by an analytic function g_t (z). Because this preserves the real axis outside K_t it is in fact real analytic. It is not unique, but can be made so by imposing the behaviour as z → ∞

It can be shown that, as the path grows, the coefficient of 1/z is monotonic increasing (essentially it is the electric dipole moment of K_t and its mirror image in the real axis). Therefore, we may reparametrise time so that this coefficient is 2t. (The factor 2 is conventional.) Note that the length of the curve is not be a useful parametrisation in the continuum limit, since the curve is a fractal.

The function g_t (z) maps the whole boundary of K_t onto part of the real axis. In particular, it maps the growing tip τ_t to a real point a_t. Any point on the real axis outside K_t remains on the real axis. As the path grows, the point a_t moves on the real axis. For the path to describe a curve, it must always grow only at its tip, and this means that the function a_t must be continuous, but not necessarily differentiable.

A simple but instructive example is when γ is a straight line growing vertically upwards from a fixed point a. In this case

This satisfies (15), and . More complicated deterministic examples can be found [18]. In particular, a_t t^1/2 describes a straight line growing at a fixed angle to the real axis.

Loewner’s idea [4] was to describe the path γ and the evolution of the tip τ_t in terms of the evolution of the conformal mapping g_t (z). It turns out that the equation of motion for g_t (z) is simple

This is Loewner’s equation. The idea of the proof is straightforward. Imagine evolving the path for a time t, and then for a further short time δt. The image of K_{t + δt} under g_t is a short vertical line above the point a_t on the real axis. Thus, we can write, using (16)

Differentiating with respect to δt and then letting δt → 0, we obtain (17).

Note that, even if a_t is not differentiable (as is the case for SLE), (17) gives for each point z₀ a solution g_t (z₀) which is differentiable with respect to t, up to the time when g_t (z₀) = a_t. This is the time when z₀ is first included in K_t. However, it is sometimes (see Section 5) useful to normalise the Loewner mapping differently, defining gˆ_t (z) = g_t (z) − a_t, which always maps the growing tip τ_t to the origin. If a_t is not differentiable, neither is gˆ_t, and the Loewner equation should be written in differential form as dgˆ_t = (2dt/gˆ_t) − da_t.

Given a growing path, we can determine the hull K_t and hence, in principle, the function g_t (z) and thereby a_t = g_t (τ_t). Conversely, given a_t we can integrate (17) to find g_t (z) and hence in determine the curve (although proving that this inverse problem gives a curve is not easy).

3.3. Schramm–Loewner evolution

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

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.

3.4. Simple properties of SLE

3.4.1. Phases of SLE

Many of the results discussed in this section have been proved by Rohde and Schramm [19]. First, we address the question of how the trace (the trajectory of τ_t) looks for different values of κ. For κ = 0, it is a vertical straight line. As κ increases, the trace should randomly turn to the L or R more frequently. However, it turns out that there are qualitative differences at critical values of κ. To see this, let us first study the process on the real axis. Let x_t = g_t (x₀) − a_t be the distance between the image at time t of a point which starts at x₀ and the image a_t of the tip. It obeys the stochastic equation

Physicists often write such an equation as where η_t is ‘white noise’ of strength κ. Of course this does not make sense since x_t is not differentiable. Such equations are always to be interpreted in the ‘Ito sense,’ that is, as the limit as δt → 0 of the forward difference equation .

Eq. (20) is known as the Bessel process. (If we set R_t = (D − 1)^1/2x_t/2 and κ² = 4/(D − 1) it describes the distance R_t from the origin of a Brownian particle in D dimensions.) The point x_t is repelled from the origin but it is also subject to a random force. Its ultimate fate can be inferred from the following crude argument: if we ignore the random force, , while, in the absence of the repulsive term, . Thus, for κ < 4 the repulsive force wins and the particle escapes to infinity, while for κ > 4 the noise dominates and the particle collides with the origin in finite time (at which point the equation breaks down). A more careful analysis confirms this. What does this collision signify (see Fig. 5) in terms of the behaviour of the trace? In Fig. 6, we show a trace which is about to hit the real axis at the point x₀, thus engulfing a whole region. This is visible from infinity only through a very small opening, which means that, under g_t, it gets mapped to a very small region. In fact, as the tip τ_t approaches x₀, the size of the image of this region shrinks to zero. When the gap closes, the whole region enclosed by the trace, as well as τ_t and x₀, are mapped in to the single point a_t, which means, in particular, that x_t → 0. The above argument shows that for κ < 4 this never happens: the trace never hits the real axis (with probability 1). For the same reason, it neither hits itself. Thus, for κ < 4 the trace γ is a simple curve.

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Fig. 5. A hull evolved from a₀ for time t₁, then to infinity. The measure on the image of the rest of the curve under g_t1 is the same, according to the postulates of SLE, as a hull evolved from a_t1 to ∞.

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