Identification with lattice models

Date: 2015-10-07; view: 436.

This result allows use to make a tentative identification with the various realisations of the O (n) model described in Section 2.2. We have, using (27), n = −2 cos(4π/κ) with 2 κ 4 describing the critical point at x_c, and 4 < κ 8 corresponding to the dense phase. Some important special cases are therefore:

• κ = −2: loop-erased random walks (proven in [24]);

• κ = 8/3: self-avoiding walks, as already suggested by the restriction property, Section 3.5.2; unproven, but see [22] for many consequences;

• κ = 3: cluster boundaries in the Ising model, however, as yet unproven;

• κ = 4: BCSOS model of roughening transition (equivalent to the 4-state Potts model and the double dimer model), as yet unproven; also certain level lines of a gaussian random field and the ‘harmonic explorer' (proven in [23]); also believed to be dual to the Kosterlitz–Thouless transition in the XY model;

• κ = 6: cluster boundaries in percolation (proven in [7]);

• κ = 8: dense phase of self-avoiding walks; boundaries of uniform spanning trees (proven in [24]).

It should be noted that no lattice candidates for κ > 8, or for the dual values κ < 2, have been proposed. This possibly has to do with the fact that, for κ > 8, the SLE trace is not reversible: the law on curves from r₁ to r₂ is not the same as the law obtained by interchanging the points. Evidently, curves in equilibrium lattice models should satisfy reversibility.