# percolation probability in a hexagonal region

Suppose one takes a large hexagonal region in the tiling of the plane by unit hexagons, with $n+1$ hexagons on each side, as seen in the figure below (taken from the COMAP website) for the case $n=5$.

Starting from any corner cell (call it $C$) and proceeding cyclically, color $n$ consecutive boundary hexagons blue, the next $n$ yellow, the next $n$ blue, the next $n$ yellow, the next $n$ blue, and the next $n$ yellow. Now color the interior hexagons blue and yellow uniformly at random. This determines three (non-intersecting) percolation paths from the boundary of the region to itself, with yellow hexagons on one side of each path and blue hexagons on the other.

What is the probability, in the limit as $n \rightarrow \infty$, that the percolation path that starts next to $C$ will terminate at the diametrically opposite point on the boundary of the region?

I know that Smirnov et al. have proved theorems asserting conformal invariance for percolation on this lattice, but I don't know the technical details, so I don't know if all the hypotheses that those theorems require are satisfied here. Assuming the answer is "yes", then the question I'm asking is a special case of a much more general (and natural) question about a probability model associated with pairings of $2n$ points on the boundary of a disk, and I'd like to learn more about that question as well (though for my purposes, it really is the case $n=3$ with 6 equally-spaced points that I want to know about, and a closed-form expression or close approximation to the probability in question).

[Added after j.c.'s proposed solution: Note that the probability is less than 1/3, since (labeling the corners 1,2,3,4,5,6 in cyclic order) the events 1-is-connected-to-4, 2-is-connected-to-5, and 3-is-connected-to-6 are disjoint, have equal probability (by symmetry), and have total probability less than 1. So .4295722 is not a possible answer.]

The papers "Euler integrals for commuting SLEs" by Julien Dubédat and "Logarithmic operator intervals in the boundary theory of critical percolation" by Jacob JH Simmons contain formulas and equations satisfied by probabilities for different crossing events in hexagons for continuum percolation at criticality. It was not immediately clear to me if due to Smirnov's proof of Cardy's formula these formulas are rigorously proven or if there are still some missing steps.

Here is a picture of the 5 different crossing configurations (from Simmons's paper).

I will work off of this figure in what follows. Let the top corner of the hexagon (adjacent to the blue-colored side B) be your corner "C", then the event that the percolation interface terminates at the bottom corner is $W^{BC}_A$. (A previous edit of this answer was incorrect as I had misread the question).

Dubédat and Simmons have computed all of these events analytically for the case of a regular hexagon and they are (eqs 27,28 of Simmons paper):

$W^{(ABC)}=W_{ABC}=\frac{1}{2}-\frac{3^{5/2}\Gamma(2/3)^9}{2^{10/3}\pi^5}{}_3F_2\left(1,\frac{5}{6},\frac{5}{6};\frac{3}{2}\frac{3}{2}|1\right)\approx0.288717$

The equality between events here is because $p_c=1/2$.

$W^{AB}_C=W^{BC}_A=W^{CA}_B=\frac{3^{3/2}\Gamma(2/3)^9}{2^{7/3}\pi^5}{}_3F_2\left(1,\frac{5}{6},\frac{5}{6};\frac{3}{2}\frac{3}{2}|1\right)\approx0.140855$

The equalities between events here are by rotational symmetry.

As for your more general question about $2n$-gons, Steven M Flores and Peter Kleban are currently writing up some results in that direction. See these two papers.

• Now that I've actually looked at those articles (something I should have done sooner!), I've realized that j.c.'s answer is incorrect. For one thing, it's greater than 1/3. To avoid confusion, my original post should have stressed the difference between my sort of connection events and the more traditional sort of crossing events people study. Is there a link between the two? That'd be great, but I don't see it yet. – James Propp Dec 10 '13 at 4:53
• @JamesPropp I had misread your question when I posted this, and I think your earlier comment identified what you really wanted as $W^{AB}_C$, which is 0.140855; I've now edited the answer. – j.c. Dec 10 '13 at 9:17
• $@$ j.c.: I deleted the comment about wanting $W_C^{AB}$ because I'm no longer certain that the two questions are equivalent. We should probably discuss this via the chat feature of MathOverflow (or whatever it's called) but I don't know how to initiate a chat. – James Propp Dec 10 '13 at 15:59
• I also have never used the chat feature before, but I don't mind discussing in comments. I think a lot of confusion may have stemmed from my earlier misreading of your question -- I didn't understand that by "percolation path" you meant the interface between blue and yellow, and instead I thought you were asking for the probability that a blue path between C and the other corner exists. I believe my answer is now correct. Please let me know if there is anything that is still unclear. – j.c. Dec 10 '13 at 16:08
• I found the mistake in my thinking that led me to an erroneous interpretation of $W_A^{BC}$. Now I see that this number is indeed exactly what I was asking about, so j.c. has answered my question, and the correct answer is (approximately) 0.140855. – James Propp Dec 10 '13 at 18:01