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Contest based on a directed polymer in a random medium

Clément Sire

par Clément Sire - 24 juillet 2008

We introduce a simple one-parameter game derived from a model describing the properties of a directed polymer in a random medium. At his turn, each of the two players picks a move among two alternatives in order to maximize his final score, and minimize opponent’s return. For a game of length n, we find that the probability distribution of the final score S_n develops a traveling wave form, {\rm Prob}(S_n=m)=f(m-v\,n), with the wave profile f(z) unusually decaying as a double exponential for large positive and negative z. In addition, as the only parameter in the game is varied, we find a transition where one player is able to get his maximum theoretical score. By extending this model, we suggest that the front velocity v is selected by the nonlinear marginal stability mechanism arising in some traveling wave problems for which the profile decays exponentially, and for which standard traveling wave theory applies.