Winding angle distribution

Date: 2015-10-07; view: 370.

A simple property which can be inferred from the Coulomb gas formulation is the winding angle distribution. Consider a cylinder of circumference 2π and a path that winds around it. What the probability that it winds through an angle θ around the cylinder while it moves a distance L 1 along the axis? This will correspond to a height difference Δh = π(θ/2π) between the ends of the cylinder, and therefore an additional free energy (g/4π)(2πL)(θ/2L)². The probability density is therefore

P(θ) exp(-gθ2/8L),

(11)

so that θ is normally distributed with variance (4/g)L. This result will be useful later (Section 3.6) for comparison with SLE.

2.4.2. N-leg exponent

As a final simple exponent prediction, consider the correlation function Φ_N (r₁)Φ_N (r₂) of the N-leg operator, which in the language of the O (n) model is Φ_N=s_a1,…,s_aN, where none of the indices are equal. It gives the probability that N mutually non-intersecting curves connect the two points. Taking them a distance L ℓ apart along the cylinder, we can choose to orient them all in the same sense, corresponding to a discontinuity in h of Nπ in going around the cylinder. Thus, we can write , where 0 v < ℓ is the coordinate around the cylinder, and . This gives

ΦN(r1)ΦN(r2) exp(-(g/4π)(Nπ/ℓ)2L+(πL/6ℓ)-(πcL/6ℓ)).

(12)

The second term in the exponent comes from the integral over the fluctuations , and the last from the partition function. They differ because in the numerator, once there are curves spanning the length of the cylinder, loops around it, which give the correction term in (10), are forbidden. Eq. (12) then gives