https://doi.org/10.1140/epjb/e2006-00341-4
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 garel@spht.saclay.cea.fr
Received:
7
June
2006
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