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 $R^2\sim L$ and by free-energy fluctuations of order $\Delta F(L) \sim O(1)$. The low-temperature phase is characterized by an anomalous wandering exponent $R^2\mathrm{/}L\sim L^{\omega }$ and by free-energy fluctuations of order $\mathrm{\Delta }F(L)\sim L^{\omega }$ where $\omega \sim 0.18$. In this paper, we first study the scaling behavior of various properties to localize the critical temperature $T_c$. Our results concerning $R^2\mathrm{/}L$ and $\mathrm{\Delta }F(L)$ point towards $0.76<T_c\le T_2=0.79$, 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 $\overline{Z_L^2}\mathrm{/}(\overline{Z_L}{)}^2$ remains finite as $L\to \mathrm{\infty }$). We then present histograms for the free-energy, energy and entropy over disorder samples. For $T\gg T_c$, the free-energy distribution is found to be Gaussian. For $T \ll T_c$, 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 $\mathrm{\Delta }S\sim L^{1\mathrm{/}2}$ and follow a Gaussian distribution, in agreement with the droplet predictions, where the free-energy term $\mathrm{\Delta }F\sim L^{\omega }$ is a near cancellation of energy and entropy contributions of order $L^{1\mathrm{/}2}$.