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A remark on diffusion of directed polymers in random environments

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We consider a system of random walks or directed polymers interacting with an environment which is random in space and time. Under minimal assumptions on the distribution of the environment, we prove that this system has diffusive behavior with probability one ifd>2 and β<β0, where β0 is defined in terms of the probability that the symmetric nearest neighbor random walk on thed-dimensional integer lattice ever returns to its starting point. We also obtain a precise estimate for the mean square displacement of this system.

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Communicated by J. L. Lebowitz

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Song, R., Zhou, X.Y. A remark on diffusion of directed polymers in random environments. J Stat Phys 85, 277–289 (1996). https://doi.org/10.1007/BF02175566

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Key Words

  • Random walks
  • directed polymers
  • random environment
  • martingales