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• n = −2 corresponds to the loop-erased random walk. This is an ordinary random walk in which every loop, once it is formed, is erased. Taking n = −2 in the O (n) model of non-intersecting loops has this effect.
2.3. Potts model
Another important model which may described in terms of random curves in the Q-state Potts model. This is most easily considered on square lattice, at each site of which is a variable s (r) which can take Q (initially a positive integer) different values. The partition function is
| | (6) |
with eβJ = (1 − p)−1. The product may be expanded in a similar way to the case of the Ising model. All possible graphs 
will appear. Within each connected component of the Potts spins must be equal, giving rise to a factor Q when the trace is performed. The result is
| | (7) |
where 
is the number of edges in , 
is the number in its complement, and 
is the number of connected components of , which are called Fortuin–Kasteleyn (FK) clusters. This is the random cluster representation of the Potts model. When p is small, the mean cluster size is small. As p → pc, it diverges, and for p > pc there is an infinite cluster which contains a finite fraction of all the sites in the lattice. It should be noted that these FK clusters are not the same as the spin clusters within which the original Potts spins all take the same value.
The limit Q → 1 gives another realisation of percolation—this time bond percolation on the square lattice. For Q → 0 there is only one cluster. If at the same time x → 0 suitably, all loops are suppressed and the only graphs which contribute are spanning trees, which contain every site of the lattice. In the Potts partition function each possible spanning tree is counted with the same weight, corresponding to the problem of uniform spanning trees (UST). The ensemble of paths on USTs connecting two points r1 and r2 turns out to be be that of loop-erased random walks.
The random cluster model may be realised as a gas of dense loops in the way illustrated in Fig. 3. These loops lie on the medial lattice, which is also square but has twice the number of sites. It may be shown that, at pc, the weights for the clusters are equivalent to counting each loop with a fugacity 
. Thus, the boundaries of the critical FK clusters in the Q-state Potts model are the same in the scaling limit (if it exists) as the closed loops of the dense phase of the O (n) model, with 
.
(8K)
Fig. 3. Example of FK clusters (heavy lines) in the random cluster representation of the Potts model, and the corresponding set of dense loops (medium heavy) on the medial lattice. The loops never cross the edges connecting sites in the same cluster.
To generate an open path in the random cluster model connecting sites r1 and r2 on the boundary we must choose ‘wired’ boundary conditions, in which p = 1 on all the edges parallel to the boundary, from r1 to r2, and free boundary conditions, with p = 0, along the remainder.
2.4. Coulomb gas methods
Many important results concerning the O (n) model can be derived in a non-rigorous fashion using so-called Coulomb gas methods. For the purposes of comparison with later results from SLE, we now summarise these methods and collect a few relevant formulae. A much more complete discussion may be found in the review by Nienhuis [3].
We assume that the boundary conditions on the O (n) spins are free, so that the partition function is a sum over closed loops only. First orient each loop at random. Rather than giving clockwise and anti-clockwise orientations the same weight n/2, give them complex weights e±6iχ, where n = e6iχ + e−6iχ = 2 cos 6χ. These may be taken into account, on the honeycomb lattice, by assigning a weight e±iχ at each vertex where an oriented loop turns R (respectively, L). This transforms the non-local factors of n into local (albeit complex) weights depending only on the local configuration at each vertex.
Next transform to the height variables described above. By convention, the heights are taken to be integer multiples of π. The local weights at each vertex now depend only on the differences of the three adjacent heights. The crucial assumption of the Coulomb gas approach is that, under the RG, this model flows to one in which the lattice can be replaced by a continuum, and the heights go over into a gaussian free field, with partition function Z = ∫ e−S [h] [dh], where
| | (8) |
As it stands, this is a simple free field theory. The height fluctuations grow logarithmically: 
(h (r1) − h (r2))2
(2/g) ln|r1 − r2|, and the correlators of exponentials of the height decay with power laws
|
| (9) |
where xq = q2/2g. All the subtleties come from the combined effects of the phase factors and the boundaries or the topology. This is particularly easy to see if we consider the model on a cylinder of circumference ℓ and length L 
ℓ. In the simple gaussian model (8) the correlation function between two points a distance L apart along the cylinder decays as exp (−2πxqL/ℓ). However, if χ ≠ 0, loops which wrap around the cylinder are not counted correctly by the above prescription, because the total number of left turns minus right turns is then zero. We may arrange the correct factors by inserting e±6iχh/π at either end of the cylinder. This has the effect of modifying the partition function: one finds ln Z (πc/6) (L/ℓ) with
| | (10) |
This dependence of the partition function is one way of determining the so-called central charge of the corresponding CFT (Section 5). The charges at each end of the cylinder also modify the scaling dimension xq to (1/2g)((q − 6iχ/π)2 − (6iχ/π)2).
The value of g may be fixed [17] in terms of the original discreteness of the height variables as follows: adding a term −λ∫ cos 2hd2r to S in (8) ensures that, in the limit λ → ∞, h will be an integer multiple of π. For this deformation not to affect the critical behaviour, it must be marginal in the RG sense, which means that it must have scaling dimension x2 = 2. This condition then determines g = 1 − 6χ/π.
2.4.1. Winding angle distribution
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) | (11) |
exp(-gθ2/8L),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 ΦN (r1)ΦN (r2) of the N-leg operator, which in the language of the O (n) model is ΦN=sa1,…,saN, 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
|
| (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
| xN=(gN2/8)-(g-1)2/2g. | (13) |
3. SLE
3.1. The postulates of SLE
SLE gives a description of the continuum limit of the lattice curves connecting two points on the boundary of a domain 
which were introduced in Section 2. The idea is to define a measure 
on these continuous curves. (Note that the notion of a probability density of such objects does not make sense, but the more general concept of a measure does.)
There are two basic properties of this continuum limit which must either be assumed, or, better, proven to hold for a particular lattice model. The first is the continuum version of Property 3.1:
Property 3.1 Continuum version
Denote the curve by γ, and divide it into two disjoint parts: γ1 from r1 to τ, and γ2 from τ to r2. Then the conditional measure 
is the same as 
.
This property we expect to be true for the scaling limit of all such curves in the O(n) model (at least for n 
0), even away from the critical point. However, the second property encodes the notion of conformal invariance, and it should be valid, if at all, only at x = xc and, separately, for x > xc.
Property 3.2 Conformal invariance
Let Φ be a conformal mapping of the interior of the domain onto the interior of 
, so that the points (r1, r2) on the boundary of are mapped to points 
on the boundary of . The measure μ on curves in induces a measure Φ 
μ on the image curves in . The conformal invariance property states that this is the same as the measure which would be obtained as the continuum limit of lattice curves from 
to 
in . That is
| | (14) |
3.2. Loewner’s equation
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