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Suppose we have a shape bounded by a simple closed curve $\gamma \subset \mathbb{C}$, with points $A,B,C,D$ in cyclic order on the curve.

If we randomly color the interior of that shape in half red and half blue, what is the probability of a blue path from $AB$ to $CD$?

This turns out to depend on only one parameter, a generalization of the familiar cross-ratio, and we're looking for references on it.

Given four points $A,B,C,D$ on a circle, the cross-ratio is defined as: $$[A,B,C,D] = -\dfrac{(a - b)(c - d)}{(b - c)(d - a)}$$ where $a,b,c,d \in \mathbb{C}$ are the complex coordinates of the points.

Given points $A, B, C, D \in \gamma$, we can define their cross-ratio $\chi$ along $\gamma$ as the complex cross-ratio $[f(A), f(B), f(C), f(D)]$ of their images, where $f$ is any conformal equivalence from the interior of $\gamma$ to the open unit disc. The Riemann mapping theorem shows that there is such a conformal equivalence, unique up to a Moebius transformation of the output disc, and the cross ratio is invariant under those Moebius transformations.

We define the percolation probability by considering a hexagonal honeycomb of edge-length $\epsilon$ restricted to the interior of the closed curve, and randomly colouring each hexagon either red or blue independently with probability $\frac{1}{2}$. Let $p$ denote the limiting probability (as $\epsilon \rightarrow 0$) that a blue path exists from $AB$ to $CD$.

Then conformality gives us the existence of some monotone-increasing function $F : (0, \infty) \rightarrow (0, 1)$, independent of $\gamma$, such that $p \equiv F(\chi)$. In fact $F$ has a nice closed form in terms of hypergeometric functions, derived by conformally mapping the unit disc to an equilateral triangle with a Schwarz-Christoffel transformation m, and appealing to "Cardy's formula" in Grimmett's Percolation on Graphs book.

Question: Has this generalisation of cross-ratio been explored before?

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    $\begingroup$ I edited to start from the modern concerns, hoping to raise the profile of the question above the classical cross-ratio. $\endgroup$ – Matt F. Feb 24 '17 at 3:02

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