Numerical study of a directed polymer in a 1+3 dimensional random medium
Résumé
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 R2 ∼L and by free-energy fluctuations of order ΔF(L) ∼O(1). The low-temperature phase is characterized by an anomalous wandering exponent R2/L ∼Lω and by free-energy fluctuations of order ΔF(L) ∼Lω where ω∼0.18. In this paper, we first study the scaling behavior of various properties to localize the critical temperature Tc. Our results concerning R2/L and ΔF(L) point towards 0.76 < Tc ≤T2=0.79, so our conclusion is that Tc is equal or very close to the upper bound T2 derived by Derrida and coworkers (T2 corresponds to the temperature above which the ratio Z2L¯/(ZL¯)2 remains finite as L ↦ ∞). We then present histograms for the free-energy, energy and entropy over disorder samples. For T ≫Tc, the free-energy distribution is found to be Gaussian. For T ≪Tc, 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 ∼L1/2 and follow a Gaussian distribution, in agreement with the droplet predictions, where the free-energy term ΔF ∼Lω is a near cancellation of energy and entropy contributions of order L1/2.
