In percolation in an L X L box, there is a simple algorithm which
decides whether or not there is a path connecting opposite sides of
the box after looking at order L^{7/4} sites. A lower bound is given
by the minimum path length, which has been measured by Grassberger to
be order L^{1.131}. We study a random-turn version of the classical
game of Hex and determine the optimal strategy, which leads to an
algorithm for the percolation question that appears to take order
L^{1.6} queries. Joint work with Yuval Peres, Oded Schramm, and Scott
Sheffield.

Lecture #10817

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