Consider the critical percolation problem on the hexagonal lattice: each of (tiny) hexagons is independently declared «open» or «closed» with probability (1/2) — by a fair coin tossing. Assume that on the boundary of a simply connected domain four points A,B,C,D are marked. Then either there exists an «open» path, joining AB and CD, or there is a «closed» path, joining AD and BC (one can recall the famous «Hex» game here). Cardy’s formula, rigorously proved by S. Smirnov, gives an explicit value of the limit of such percolation probability, when the same smooth domain is put onto lattices with smaller and smaller mesh. Though, a next natural question is: what if more than four points are marked? And thus that there are more possible configurations of open/closed paths joining the arcs? 

In our joint work with M. Khristoforov we obtain the answer as an explicit integral for the case of six marked points on the boundary, passing through Fuchsian differential equations, Riemann surfaces, and Riemann-Hilbert problem. We also obtain a generalization of this answer to the case when one of the marked points is inside the domain (and not on the boundary).