Contest based on a directed polymer in a random medium.
We introduce a simple one-parameter game derived from a model describing the properties of a directed polymer in a random medium. At its turn, each of the two players picks a move among two alternatives in order to maximize its final score, and minimize the 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, Prob(S_{n}=m)=f(m-vn) , with the wave profile f(z) decaying unusually 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 its 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.
Sire
Clément
C
Laboratoire de Physique Théorique-IRSAMC, UPS and CNRS, Université de Toulouse, 31062 Toulouse, France. clement.sire@irsamc.ups-tlse.fr