Crossing exponent

Date: 2015-10-07; view: 514.

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

(36)

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

(37)

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