Mean field theory of directed polymers in a random medium and beyond

Starting from the mean field theory of directed polymers in a random medium, one can obtain 1/d expansions, valid for hypercubic lattices in the limit d → ∞. For finite dimensional lattices, one can predict bounds for the transition temperature and show that the specific heat exponent α is always negative (α ≤ 0). The mean field theory can be extended to the case where the statistical weights of the walks are no longer all positive, giving rise to interference effects. Lastly the problem of the branched polymers in a random medium is discussed.

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