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Numerical study of the directed polymer in a 1+3 dimensional random medium | | Cecile Monthus
; Thomas Garel
; | | Date: |
6 Jun 2006 | | Subject: | Disordered Systems and Neural Networks | | Abstract: | 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 $R^2/L sim L^{omega}$ and by free-energy fluctuations of order $Delta F(L) sim L^{omega}$ 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 $R^2/L$ and $Delta F(L)$ point towards $0.76 < T_c leq T_2=0.79$, 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 $ar{Z_L^2}/(ar{Z_L})^2$ remains finite as $L o infty$). 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 $Delta S sim L^{1/2}$ and follow a Gaussian distribution, in agreement with the droplet predictions, where the free-energy term $Delta F sim L^{omega}$ is a near cancellation of energy and entropy contributions of order $L^{1/2}$. | | Source: | arXiv, cond-mat/0606132 | | Services: | Forum | Review | PDF | Favorites | |
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