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<Superscript>2</Superscript> ∼L and by free-energy fluctuations of order ΔF(L) ∼O(1). The low-temperature phase is characterized by an anomalous wandering exponent R<Superscript>2</Superscript>/L ∼L<Superscript>ω</Superscript> and by free-energy fluctuations of order ΔF(L) ∼L<Superscript>ω</Superscript> where ω∼0.18. In this paper, we first study the scaling behavior of various properties to localize the critical temperature T<Subscript>c</Subscript>. Our results concerning R<Superscript>2</Superscript>/L and ΔF(L) point towards 0.76 > T<Subscript>c</Subscript> ≤T<Subscript>2</Subscript>=0.79, so our conclusion is that T<Subscript>c</Subscript> is equal or very close to the upper bound T<Subscript>2</Subscript> derived by Derrida and coworkers (T<Subscript>2</Subscript> corresponds to the temperature above which the ratio <InlineEquation ID="Equ1"> <EquationSource Format="TEX">$\bar{Z_L^2}/(\bar{Z_L})^2$</EquationSource> </InlineEquation> remains finite as L ↦ ∞). We then present histograms for the free-energy, energy and entropy over disorder samples. For T ≫T<Subscript>c</Subscript>, the free-energy distribution is found to be Gaussian. For T ≪T<Subscript>c</Subscript>, 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<Superscript>1/2</Superscript> and follow a Gaussian distribution, in agreement with the droplet predictions, where the free-energy term ΔF ∼L<Superscript>ω</Superscript> is a near cancellation of energy and entropy contributions of order L<Superscript>1/2</Superscript>. Copyright EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2006