Numerical study of the directed polymer in a 1 + 3 dimensional random medium
Service de Physique Théorique, CEA/DSM/SPhT, Unité de recherche associée au CNRS, 91191 Gif-sur-Yvette Cedex, France
Corresponding author: a email@example.com
Published online: 6 September 2006
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 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.
PACS: 02.50.-r – Probability theory, stochastic processes, and statistics / 64.70.-p – Specific phase transitions / 65.60.+a – hermal properties of amorphous solids and glasses: heat capacity, thermal expansion, etc. / 05.40.-a – Fluctuation phenomena, random processes, noise, and Brownian motion
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2006