Probing the tails of the ground-state energy distribution for the directed polymer in a random medium of dimension d=1,2,3 via a Monte Carlo procedure in the disorder.
In order to probe with high precision the tails of the ground-state energy distribution of disordered spin systems, Körner, Katzgraber, and Hartmann have recently proposed an importance-sampling Monte Carlo Markov chain in the disorder. In this paper, we combine their Monte Carlo procedure in the disorder with exact transfer matrix calculations in each sample to measure the negative tail of ground-state energy distribution Pd(E0) for the directed polymer in a random medium of dimension d=1,2,3. In d=1, we check the validity of the algorithm by a direct comparison with the exact result, namely, the Tracy-Widom distribution. In dimensions d=2 and d=3, we measure the negative tail up to ten standard deviations, which correspond to probabilities of order Pd(E0) approximately 10(-22). Our results are in agreement with Zhang's argument, stating that the negative tail exponent eta(d) of the asymptotic behavior lnPd(E0) approximately -|E0|eta(d) as E0-->-infinity is directly related to the fluctuation exponent theta(d) [which governs the fluctuations DeltaE0(L) approximately Ltheta(d) of the ground-state energy E0 for polymers of length L] via the simple formula eta(d)=1/[1-theta(d)]. Throughout the paper, we comment on the similarities and differences with spin glasses.
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Service de Physique Théorique, CEA/DSM/SPhT, Unité de recherche associée au CNRS, 91191 Gif-sur-Yvette cedex, France.
Garel
Thomas
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D003102:Colloids / Q000737:chemistry*; D003198:Computer Simulation; D004548:Elasticity; D004735:Energy Transfer; D008956:Models, Chemical; D008958:Models, Molecular; D015233:Models, Statistical; D008968:Molecular Conformation; D009010:Monte Carlo Method; D044367:Phase Transition; D011108:Polymers / Q000737:chemistry*; D012997:Solvents / Q000737:chemistry*; D016008:Statistical Distributions
D003102:Colloids; D011108:Polymers; D012997:Solvents