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Percolation in a triangle on a square lattice

Percolation on a plane is usually associated with clusters spanning two opposite sides of a rectangular system. Here we investigate three-leg clusters generated on a square lattice and spanning the three sides of equilateral triangles. If the position and orientation of the triangles relative to the lattice are uniformly randomized, one obtains an efficient method of determining the percolation threshold, on par with the most advanced Monte Carlo methods developed for the rectangular geometry. The universal crossing probability for three-leg clusters is geometry-independent, which opens a way for further improvements of the method.