The scaling limit of the spin cluster boundaries of the Ising model with domain wall boundary conditions is SLE with κ = 3. We hypothesise that the three-state Potts model with appropriate boundary conditions has spin cluster boundaries which are also SLE in the scaling limit, but with κ = 10/3. To test this, we generate samples using the Wolff algorithm and test them against predictions of SLE: we examine the statistics of the Loewner driving function, estimate the fractal dimension and test against Schramm's formula. The results are in support of our hypothesis.

The probability that a point is to one side of a curve in SchrammLoewner evolution (SLEκ) can be obtained alternatively using boundary conformal field theory (BCFT). We extend the BCFT approach to treat two curves, forming, for example, the left and right boundaries of a cluster. This proves to correspond to a generalisation to SLE(κ, ρ), with ρ = 2. We derive the probabilities that a given point lies between two curves or to one side of both. We find analytic solutions for the cases κ = 0, 2, 4, 8/3, 8. The result for κ = 6 leads to predictions for the current distribution at the plateau transition in the semiclassical approximation to the quantum Hall effect.

The O(n) spin model in two dimensions may equivalently be formulated as a loop model, and then mapped to a height model which is conjectured to flow under the renormalization group to a conformal field theory (CFT). At the critical point, the order n terms in the partition function and correlation functions describe single self-avoiding loops. We investigate the ensemble of these self-avoiding loops using twist operators, which count loops which wind non-trivially around them with a factor −1. These turn out to have level two null states and hence their correlators satisfy a set of partial differential equations. We show that partlyconnected parts of the four point function count the expected number of loops which separate one pair of points from the other pair, and find an explicit expression for this. We argue that the differential equation satisfied by these expectation values should have an interpretation in terms of a stochastic(Schramm)-Loewner evolution (SLEκ) process with κ = 6. The two point function in a simply connected domain satisfies a closely related set of equations. We solve these and hence calculate the expected number of single loops which separate both points from the boundary.

This work concerns the dynamical two-point spin correlation functions of the transverse Ising quantum chain at finite (non-zero) temperature, in the universal region near the quantum critical point. They are correlation functions of twist fields in the massive Majorana fermion quantum field theory. At finite temperature, these are known to satisfy a set of integrable partial differential equations, including the sinh-Gordon equation. We apply the classical inverse scattering method to study them, finding that the "initial scattering data" corresponding to the correlation functions are simply related to the one-particle finite-temperature form factors calculated recently by one of the authors. The set of linear integral equations (Gelfand-Levitan-Marchenko equations) associated to the inverse scattering problem then gives, in principle, the two-point functions at all space and time separations, and all temperatures. From them, we evaluate the large-time asymptotic expansion "near the light cone", in the region where the difference between the space and time separations is of the order of the correlation length.