Numerical study of the directed polymer in a 1 + 3 dimensional random medium

The directed polymer in a 1+3 dimensional random medium is known to present a disorder-induced phase transition. For a polymer of length L, the high temperature phase is characterized by a diffusive behavior for the end-point displacement R 2 ∼L and by free-energy fluctuations of order ΔF(L) ∼O(1). The low-temperature phase is characterized by an anomalous wandering exponent R 2 /L ∼L ω and by free-energy fluctuations of order ΔF(L) ∼L ω where ω∼0.18. In this paper, we first study the scaling behavior of various properties to localize the critical temperature T c . Our results concerning R 2 /L and ΔF(L) point towards 0.76 > T c ≤T 2 =0.79, so our conclusion is that T c is equal or very close to the upper bound T 2 derived by Derrida and coworkers (T 2 corresponds to the temperature above which the ratio $\bar{Z_L^2}/(\bar{Z_L})^2$ remains finite as L ↦ ∞). We then present histograms for the free-energy, energy and entropy over disorder samples. For T ≫T c , the free-energy distribution is found to be Gaussian. For T ≪T c , the free-energy distribution coincides with the ground state energy distribution, in agreement with the zero-temperature fixed point picture. Moreover the entropy fluctuations are of order ΔS ∼L 1/2 and follow a Gaussian distribution, in agreement with the droplet predictions, where the free-energy term ΔF ∼L ω is a near cancellation of energy and entropy contributions of order L 1/2 . Copyright EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2006