SLE (2) : Property of locality

One of the interest to study $SLE_{\kappa}$ is that it simulates the critical behaviors of a lot of random models such as self-avoiding random walk, the interface of Ising and uniform spanning tree. Here, we give an example of the locality property of $SLE_{6}$ and its connection of Ising model in critical case.

Site Ising model

We first introduce the Ising model. This is so called site Ising model that each site has $\frac{1}{2}$ probability to be black or white. We suppose that the negative axis is black and the positive axis is white and a process starts from $0$ who only expolres the  interface of the two. It has two properties:

1. Conformal invariance. It is believed that after an conformal mapping, the process follow the same property.
2. Domain Markov property. When the diameter of lattice goes to $0$, the process should be independent to the one already explored since the trace is determined locally.

One has a third property called locality : that a process in domain has the same law before it goes out of the domain, since also the trace is determined locally.

The conformal invariance and domain Markov implies that its scaling limit should be a $SLE_{\kappa}$.

$SLE_6$ and locality

But why this $SLE_{\kappa}$ should be $SLE_{6}$?

That is required by the property of locality. Generally, we can define the conformal  image of $SLE_{\kappa}$ in domain $D$,  but it doesn't follow the same law as a random process. For example, we define a mapping $$ \phi : D \rightarrow \mathbb{H} $$, then the equation becomes $$\partial_t \tilde{g}_t (z) = \frac{\partial_t hcap}{\partial_t \tilde{g}_t (z) - \tilde{U}_t}$$

Here $\tilde{U}_t$ is not necessarily a standard Brownian motion and $\kappa = 6$ is the case to make it as a real one.

We skip the technique calculus but just talks about its sense. It means that before going out of the boundary, the image $\phi(K_t)$ follows also a $SLE_6$ process. This one implies the property of locality. 

Cardy formula

The most interesting story about the $SLE_6$ is its behavior one equilateral triangle. We first calculate one probability of $SLE_6$ on $\mathbb{H}$ : $\mathbb{P}(\tau_{-y} < \tau_{1}) = F(\frac{y}{1+y}) $ where $$F(z) = c^{-1}\int_0^z \frac{1}{u^{\frac{2}{3}} (1-u)^{\frac{2}{3}}} du$$

which coincides with  the Schwartz-Christoffel formula in complex analysis and satisfies $$F(1+y) = 1 + e^{2\pi i / 3} F(\frac{y}{1+y})$$ which  sends $0, 1, \infty$ to $0, 1, e^{\pi i /3}$ respectively.

We construct an other conformal mapping $\Phi : \mathbb{H} \rightarrow \triangle$ which sends $0, 1, -y$ to $0, 1, e^{\pi i /3}$ respectively. We  can show in this case, $\Phi(\infty) = F(1+y)$. So we get the property that $$\mathbb{P}(\tau_{-y} < \tau_{1}) = \frac{|\Phi(\infty) \Phi(1)|}{|\Phi(-y) \Phi(1)|}$$ 

Using the locality of $SLE_{6}$, this interprets that a $SLE_{6}$ from $0$ to $\Phi(\infty)$ in $\triangle$, the first time to hit each side just has the probability of the length. A similar argument shows that the same process from $0$ to $1$ has the  uniform distribution for the first hit. 

More generally, two $SLE_{6}$ in the same triangle, one from $0$ to $1$ and another from $0$ to $e^{\pi i/3}$, share the same law before until the first hit one the opposite side.