Abstract
We consider the following twoplayer game on a graph. A token is located at a vertex, and the players take turns to move it along an edge to a vertex that has not been visited before. A player who cannot move loses. We analyze outcomes with optimal play on percolation clusters of Euclidean lattices. On Z^2 with two different percolation parameters for odd and even sites, we prove that the game has no draws provided closed sites of one parity are sufficiently rare compared with those of the other parity (thus favoring one player). We prove this also for certain ddimensional lattices with d≥3. It is an open question whether draws can occur when the two parameters are equal. On a finite ball of Z^2, with only odd sites closed but with the external boundary consisting of even sites, we identify up to logarithmic factors a critical window for the tradeoff between the size of the ball and the percolation parameter. Outside this window, one or the other player has a decisive advantage. Our analysis of the game is intimately tied to the effect of boundary conditions on maximumcardinality matchings.
Original language  English 

Pages (fromto)  37273753 
Journal  Annals of Applied Probability 
Volume  26 
Issue number  6 
Publication status  Published  2016 
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Profiles

Professor Alexander E Holroyd
Person: Academic