Article Dans Une Revue Physical Review E : Statistical, Nonlinear, and Soft Matter Physics [2001-2015] Année : 2006

Probing the tail 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

Résumé

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)∼10−22. Our results are in agreement with Zhang’s argument, stating that the negative tail exponent η(d) of the asymptotic behavior lnPd(E0)∼−∣E0∣η(d) as E0→−∞ is directly related to the fluctuation exponent θ(d) [which governs the fluctuations ΔE0(L)∼Lθ(d) of the ground-state energy E0 for polymers of length L] via the simple formula η(d)=1∕[1−θ(d)]. Throughout the paper, we comment on the similarities and differences with spin glasses.

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hal-00172812 , version 1 (20-04-2023)

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Cecile Monthus, Thomas Garel. Probing the tail 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. Physical Review E : Statistical, Nonlinear, and Soft Matter Physics [2001-2015], 2006, 74 (5), pp.051109. ⟨10.1103/PhysRevE.74.051109⟩. ⟨hal-00172812⟩
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