The distribution of the area of a region cut out by chordal SLE?

Question

Let $\mathbb{D}$ be the unit disc. Let $a,b \in \partial \mathbb{D}$. Let $\gamma$ be a chordal $SLE_{k}$ from $a$ to $b$.

For $k \leq 4$, $\gamma$ is a simple curve, and so $\mathbb{D} \setminus \gamma$ has two components. Say that $A$ is the left-component of $\mathbb{D} \setminus \gamma$ when we traverse $\gamma$ from $a$ to $b$.

Is there anything known about the distribution of the (Euclidean) area, $|A|$, of the random set $A$? Of course this depends on $(a,b)$.

Here are some more concrete questions:

1. If $a,b$ are antipodal, then by symmetry the expected area of $A$ is $|\mathbb{D}|/2$. What is the expected area when $a, b$ are not antipodal? I know that one can apply a conformal transformation to get back the antipodal case, but it feels unlikely that such a transformation will distort equal areas in the same way. In particular, if $a$ and $b$ were close, I would be surprised if the distribution was symmetric.
2. What is $Var |A|$, as a function of $(a,b)$?

Answer 1

The expected area of $A$ is easy to compute, in principle explicitly: $$ \mathbb{E}(\text{Area}(A))=\mathbb{E}\left(\int_\mathbb{D}\mathbb{1}_{z\in A}\right)=\int_\mathbb{D}\mathbb{P}(z\in A). $$ The probability inside the integral is given by so-called Schramm's formula and can be computed by standard SLE techniques: since in the half-plane geometry, the probability that $z$ is to the left of the curve is scale invariant, it is given by $F(\arg(z))$ for some function $F$. On the other hand, conditionally on the curve up to time $t$, it is $F(\arg(g_t(z)-\sqrt{\kappa}B_t)).$ Hence the latter quantity is a martingale. Applying Ito's formula, this yields after some computations $$ \frac{1}{2}F''(\theta)+\left(1-\frac{4}{\kappa}\right)\tan(\theta)F'(\theta)=0, $$ which, together with boundary conditions $F(0)=1$, $F(\pi)=0$, gives an explicit formula for $F$ (for $\kappa=4$ we actually have $F(\theta)=1-\theta/\pi$, and for other $\kappa$ we have a hypergeometric function). Since the quantity in question is Mobius invariant, in the disc geometry it is given by $F(\text{hm}_z(ba))$, where $\text{hm}$ is the harmonic measure.

In order to compute variance in a similar way, one would need to compute two-point function. This function is in general not known: although by an argument as above one can derive some PDEs for it, they are hard to solve in a closed form.