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Diffusion of directed polymers in a strong random environment

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Abstract

We consider a system of random walks or directed polymers interacting with an environment which is random in space and time. It was shown by Imbrie and Spencer that in spatial dimensions three or above the behavior is diffusive if the directed polymer interacts weakly with the environment and if the random environment follows the Bernoulli distribution. Under the same assumption on the random environment as that of Imbrie and Spencer, we establish that in spatial dimensions four or above the behavior is still diffusive even when the directed polymer interacts strongly with the environment. More generally, we can prove that, if the random environment is bounded and if the supremum of the support of the distribution has a positive mass, then there is an integerd 0 such that in dimensions higher thand 0 the behavior of the random polymer is always diffusive.

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References

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Communicated by J. L. Lebowitz

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Olsen, P., Song, R. Diffusion of directed polymers in a strong random environment. J Stat Phys 83, 727–738 (1996). https://doi.org/10.1007/BF02183745

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Key Words

  • Random walks
  • directed polymers
  • random environment
  • martingales