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Fig. 1. Ising model on the honeycomb lattice, with loops corresponding to a term in the expansion of (1). Alternatively, these may be thought of as domain walls of an Ising model on the dual triangular lattice.

At high temperatures (βJ 1) the spins are disordered, and their correlations decay exponentially fast, while at low temperatures (βJ 1) there is long-range order: if the spins on the boundary are fixed say, to the value +1, then s (r) ≠ 0 even in the infinite volume limit. In between, there is a critical point. The conventional approach to the Ising model focuses on the behaviour of the correlation functions of the spins. In the scaling limit, they become local operators in a quantum field theory (QFT). Their correlations are power-law behaved at the critical point, which corresponds to a massless QFT, that is a conformal field theory (CFT). From this point of view (as well as exact lattice calculations) it is found that correlation functions like s (r₁)s (r₂) decay at large separations according to power laws |r₁ − r₂|^−2x: one of the aims of the theory is to obtain the values of the exponents x as well as to compute, for example, correlators depending on more than two points.

However, there is an alternative way of thinking about the partition function (1), as follows: imagine expanding out the product to obtain 2^N terms, where N is the total number of edges. Each term may be represented by a subset of edges, or graph , on the lattice, in which, if the term xs (r)s (r′) is chosen, the corresponding edge (rr′) is included in , otherwise it is not. Each site r has either 0, 1, 2, or 3 edges in . The trace over s (r) gives 1 if this number is even, and 0 if it is odd. Each surviving graph is then the union of non-intersecting closed loops (see Fig. 1). In addition, there can be open paths beginning and ending at a boundary. For the time being, we suppress these by imposing ‘free’ boundary conditions, summing over the spins on the boundary. The partition function is then

where the length is the total of all the loops in . When x is small, the mean length of a single loop is small. The critical point x_c is signalled by a divergence of this quantity. The low-temperature phase corresponds to x > x_c. While in this phase the Ising spins are ordered, and their connected correlation functions decay exponentially, the loop gas is in fact still critical, in that, for example, the probability that two points lie on the same loop has a power-law dependence on their separation. This is the dense phase.

The loops in may be viewed in another way: as domain walls for another Ising model on the dual lattice, which is a triangular lattice whose sites R lie at the centres of the hexagons of the honeycomb lattice (see Fig. 1). If the corresponding interaction strength of this dual Ising model is (βJ) , then the Boltzmann weight for creating a segment of domain wall is e^−2(βJ) . This should be equated to x = tanh (βJ) above. Thus, we see that the high-temperature regime of the dual model corresponds to low temperature in the original model, and vice versa. Infinite temperature in the dual model ((βJ) = 0) means that the dual Ising spins are independent random variables. If we colour each dual site with s (R) = +1 black, and white if s (R) = −1, we have the problem of site percolation on the triangular lattice, critical because for that problem. Thus, the curves with x = 1 correspond to percolation cluster boundaries. (In fact in the scaling limit this is believed to be true throughout the dense phase x > x_c.)

So far we have discussed only closed loops. Consider the spin–spin correlation function

where the sites r₁ and r₂ lie on the boundary. Expanding out as before, we see that the surviving graphs in the numerator each have a single edge coming into r₁ and r₂. There is therefore a single open path γ connecting these points on the boundary (which does not intersect itself nor any of the closed loops). In terms of the dual variables, such a single open curve may be realised by specifying the spins s (R) on all the dual sites on the boundary to be +1 on the part of the boundary between r₁ and r₂ (going clockwise) and −1 on the remainder. There is then a single domain wall connecting r₁ to r₂. SLE describes the continuum limit of such a curve γ.

Note that we could also choose r₂ to lie in the interior. The continuum limit of such curves is then described by radial SLE (Section 3.6).

2.1.1. Exploration process

An important property of the ensemble of curves γ on the lattice is that, instead of generating a configuration of all the s (R) and then identifying the curve, it may be constructed step-by-step as follows (see Fig. 2). Starting from r₁, at the next step it should turn R or L according to whether the spin in front of it is +1 or −1. For independent percolation, the probability of either event is 1/2, but for x < 1 it depends on the values of the spins on the boundary. Proceeding like this, the curve will grow, with all the dual sites on its immediate left taking the value +1, and those its right the value −1. The relative probabilities of the path turning R or L at a given step depend on the expectation value of the spin on the site R immediately in front of it, given the values of the spins already determined, that is, given the path up to that point. Thus, the relative probabilities that the path turns R or L are completely determined by the domain and the path up to that point. This implies the crucial.

Property 2.1 Lattice version

Let γ₁ be the part of the total path covered after a certain number of steps. Then the conditional probability distribution of the remaining part of the curve, given γ₁, is the same as the unconditional distribution of a whole curve, starting at the tip and ending at r₂ in the domain .

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Fig. 2. The exploration process for the Ising model. At each step the walk turns L or R according to the value of the spin in front of it. The relative probabilities are determined by the expectation value of this spin given the fixed spins either side of the walk up to this time. The walk never crosses itself and never gets trapped.

In the Ising model, for example, if we already know part of the domain wall, the rest of it can be considered as a complete domain wall in a new region in which the left and right sides of the existing part form part of the boundary. This means the path is a history-dependent random walk. It can be seen (Fig. 2) that when the growing tip τ approaches an earlier section of the path, it must always turn away from it: the tip never gets trapped. There is always at least one path on the lattice from the tip τ to the final point r₂.

2.2. O (n) model

The loop gas picture of the Ising and percolation models may simply be generalised by counting each closed loop with a fugacity n

This is called the O (n) model, for the reason that it gives the partition function for n-component spins s (r) = (s₁ (r), … , s_n (r)) with

where Tr s_a (r) s_b (r) = δ_ab. Following the same procedure as before we obtain the same set of closed loops (and open paths) except that, on summing over the last spin in each closed loop, we get a factor n. The model is called O (n) because of its symmetry under rotations of the spins. The version (5) makes sense only when n is a positive integer (and note that the form of the partition function is different from that of the conventional O (n) model, where the second term is exponentiated). The form in (4) is valid for general values of n, and it gives a probability measure on the loop gas for real n 0. However, the dual picture is useful only for n = 1 and n = 2 (see below). As for the case n = 1, there is a critical value x_c (n) at which the mean loop length diverges. Beyond this, there is a dense phase.

Apart from n = 1, other important physical values of n are:

• n = 2. In this case we can view each loop as being oriented in either a clockwise or anti-clockwise sense, giving it an overall weight 2. Each loop configuration then corresponds to a configuration of integer valued height variables h (R) on the dual lattice, with the convention that the nearest neighbour difference h (R′) − h (R) takes the values 0, +1 or −1 according to whether the edge crossed by RR′ is unoccupied, occupied by an edge oriented at 90° to RR′, or at −90°. (That is, the current running around each loop is the lattice curl of h.) The variables h (R) may be pictured as the local height of a crystal surface. In the low-temperature phase (small x) the surface is smooth: fluctuations of the height differences decay exponentially with separation. In the high-temperature phase it is rough: they grow logarithmically. In between is a roughening transition. It is believed that relaxing the above restriction on the height difference h (R′) − h (R) does not change the universality class, as long as large values of this difference are suppressed, for example using the weighting exp [−β (h (R′) − h (R))²]. This is the discrete Gaussian model. It is dual to a model of two-component spins with O (2) symmetry called the XY model.

• n = 0. In this case, closed loops are completely suppressed, and we have a single non-self-intersecting path connecting r₁ and r₂, weighted by its length. Thus, all paths of the same length are counted with equal weight. This is the self-avoiding walk problem, which is supposed to describe the behaviour of long flexible polymer chains. As x→x_c^-, the mean length diverges. The region x > x_c is the dense phase, corresponding to a long polymer whose length is of the order of the area of the box, so that it has finite density.

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