The fractal dimension of SLE

Date: 2015-10-07; view: 464.

Critical exponents from SLE

Many of the critical exponents which have previously been conjectured by Coulomb gas or CFT methods may be derived rigorously using SLE, once the underlying postulates are assumed or proved. However SLE describes the measure on just a single curve, and in the papers of LSW a great deal of ingenuity has gone into showing how to relate this to all the other exponents. There is not space in this article to do these justice. Instead we describe three examples which give the flavour of the arguments, which initially may appear quite unconventional compared with the more traditional approaches.

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

(34)

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.