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The fractal dimension of any geometrical object embedded in the plane can be defined roughly as follows: let N ( ) be the minimum number of small discs of radius required to cover the object. Then if N( ) ^-d_f as → 0, d_f is the fractal dimension.

One way of computing d_f for a random curve γ in the plane is to ask for the probability P (x, y, ) that a given point ζ = x + iy lies within a distance of γ. A simple scaling argument shows that if P behaves like ^δf (x, y) as → 0, then δ = 2 − d_f. We can derive a differential equation for P along the lines of the previous calculation. The only difference is that under the conformal mapping . The differential equation (written for convenience in cartesian coordinates) is

Now P is dimensionless and therefore should have the form ( /r)^2-d_f times a function of the polar angle θ. In fact, the simple ansatz P= ^2-d_fy^α(x²+y²)^β, with α + 2β = d_f − 2 satisfies the equation. [The reason this works is connected with the simple form for the correlator Φ_{2 2,1 2,1} discussed in Section 5.4.1.] This gives α = (κ − 8)²/8κ, β = (κ − 8)/2κ, and

This is correct for κ 8: otherwise there is another solution with α = β = 0 and d_f = 2. A more careful statement and proof of this result can be found in [31].

We see that the fractal dimension increases steadily from the value 1 when κ = 0 (corresponding to a straight line) to a maximum value of 2 when κ = 8. Beyond this value γ becomes space-filling: every point in the upper half plane lies on the curve.

4.3.2. Crossing exponent

Consider a critical percolation problem in the upper half plane. What is the asymptotic behaviour as r → ∞ of the probability that the interval (0, 1) on the real axis is connected to the interval (r, ∞)? We expect this to decay as some power of r. The value of this exponent may be found by taking the appropriate limit of the crossing formula (33), but instead we shall compute it directly. In order for there to be a crossing cluster, there must be two cluster boundaries which also cross between the two intervals, and which bound this cluster above and below. Denote the upper boundary by γ. Then we need to know the probability P (r) of there being another spanning curve lying between γ and (1, r), averaged over all realisations of γ. Because of the locality property, the measure on γ is independent of the existence of the lower boundary, and is given by SLE₆ conditioned not to hit the real axis along (1, r). Note that because κ > 4 it will eventually hit the real axis at some point to the right of r. For this reason we can do the computation for general κ > 4, although it gives the actual crossing exponent only if κ = 6.

Consider the behaviour of P (r) under the conformal mapping (which maps the growing tip τ_t into 0). The crossing probability should be conformally invariant and depend only on the ratio of the lengths of the two intervals, hence, by an argument which by now should be familiar

Expanding this out, remembering as usual that (dB_t)² = dt, and setting to zero the O (dt) term, we find for r 1

with the solution P (r) r^{−(κ−4)/κ} for κ > 4. Setting κ = 6 then gives the result 1/3.

4.3.3. The one-arm exponent

Consider critical lattice percolation inside some finite region (for example a disc of radius R). What is the probability that a given site (e.g., the origin) is connected to a finite segment S of the boundary? This should decay like R^−λ, where λ is sometimes called the one-arm exponent. If we try to formulate this in the continuum, we immediately run up against the problem that all clusters are fractal with dimension <2, and so the probability of any given point being in any given cluster is zero. Instead, one may ask about the probability P (r) that the cluster connected to S gets within a distance r of the origin. This should behave like (r/R)^λ. We can now set R = 1 and treat the problem using radial SLE₆.

Consider now a radial SLE_κ which starts at e^iθ₀. If κ > 4 it will continually hit the boundary. Let P (θ − θ₀, t) be the probability that the segment (θ₀, θ) of the boundary has not been swallowed by time t. Then, by considering the evolution as usual under g_dt

where dθ = cot((θ − θ₀)/2) dt and . Setting θ₀ = 0 and equating to zero the O (dt) term, we find the time-dependent differential equation

This has the form of a backwards Fokker–Plank equation.

Now, since , it is reasonable that, after time t, the SLE gets within a distance O (e^−t) of the origin. Therefore, we can interpret P as roughly the probability that the cluster connected to (0, θ) gets within a distance r e^−t of the origin. A more careful argument [32] confirms this. The boundary conditions are P (0, t) = 0 as θ → 0, and (with more difficulty) &#x2202;_θP (θ, t) = 0 at θ = 2π. The solution may then be found by inspection to be

where λ = (κ²−16)/32κ. For percolation this gives 5/48, in agreement with Coulomb gas arguments [3].

The appearance of differential operators such as that in (39) will become clear from the CFT perspective in Section 5.4.1. If instead of choosing Neuman boundary conditions at θ = 2π we impose P = 0, the same equation gives the bulk two-leg exponent x₂, which is also related to the fractal dimension by d_f = 2− x₂.

5. Relation to conformal field theory

5.1. Basics of CFT

The reader who already knows a little about CFT will have recognised the differential equations in Section 4 as being very similar in form to the BPZ equations [33] satisfied by the correlation functions of a _2,1 operator, corresponding to a highest weight representation of the Virasoro algebra with a level 2 null state.

For those readers for whom the above paragraph makes no sense, and in any case to make the argument self-contained, we first introduce the concepts of (boundary) conformal field theory (BCFT). We stress that these are heuristic in nature—they serve only as a guide to formulating the basic principles of CFT which can then be developed into a mathematically consistent theory. For a longer introduction to BCFT see [34] and, for a complete account of CFT [35].

We have at the back of our minds a euclidean field theory defined as a path integral over some set of fundamental fields {ψ (r)}. The partition function is Z = ∫ e^−S({ψ})[dψ] where the action is an integral over a local lagrangian density. These fields may be thought of as smeared-out continuum versions of the lattice degrees of freedom. As in any field theory, this continuum limit involves renormalisation. There are so-called local scaling operators which are particular functionals of the fundamental degrees of freedom, which have the property that we can define renormalised scaling operators whose correlators are finite in the continuum limit a → 0, that is

exists. The numbers x_j are called the scaling dimensions, and are related to the various critical exponents. They are related to the conformal weights by ; the difference is called the spin of _j, and describes its behaviour under rotations. There are also boundary operators, localised on the boundary, which have only a single conformal weight equal to their scaling dimension.

The theory is developed independently of any particular set of fundamental fields or lagrangian. An important role in this is played by the stress tensor T^μν (r), defined as the local response of the action to a change in the metric:

Invariance under local rotations and scale transformations usually implies that T^μν is symmetric and traceless: . This also implies invariance under conformal transformations, corresponding to δg_μν f (r)g_μν.

In two-dimensional flat space, infinitesimal coordinate transformations r^μ→r^′μ=r^μ+α^μ(r) correspond to infinitesimal transformations of the metric with δg^μν = −(∂^μα^μ + ∂^να^ν). It is important to realise that under these transformations the underlying lattice, or UV cut-off, is not transformed. Otherwise they would amount to a trivial reparametrisation. For a conformal transformation, α^μ (r) is given by an analytic function: in complex coordinates , , so α ^z ≡ α (z) is holomorphic. However, such a function cannot be small everywhere (unless it is constant), so it is necessary to consider coordinate transformations which are not everywhere conformal.

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