The postulates of SLE

Date: 2015-10-07; view: 380.

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 = x_c and, separately, for x > x_c.

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

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