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.
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.
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通过无序蒙特卡罗程序探测维度 d=1,2,3 的随机介质中定向聚合物的基态能量分布的尾部。
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发表时间：
2006
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通讯作者：
T. Garel
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作者：
C. Monthus;T. Garel
为了高精度地探测无序自旋系统基态能量分布的尾部，Körner、Katzgraber 和 Hartmann 最近提出了无序中的重要采样蒙特卡洛马尔可夫链。在本文中，我们将无序中的蒙特卡罗程序与每个样本中的精确传递矩阵计算相结合，以测量维度 d=1,2,3 的随机介质中定向聚合物的基态能量分布 Pd(E0) 的负尾部。在 d=1 时，我们通过与精确结果（即 Tracy-Widom 分布）的直接比较来检查算法的有效性。在维度 d=2 和 d=3 中，我们测量负尾高达 10 个标准差，这对应于大约 10(-22) 的 Pd(E0) 阶概率。我们的结果与张的论点一致，指出渐近行为 lnPd(E0) 的负尾指数 eta(d) 近似 -|E0|eta(d) 作为 E0-->-无穷大 直接与涨落指数 theta(d) [通过简单的公式控制长度 L 的聚合物基态能量 E0 的涨落 DeltaE0(L) 近似 Ltheta(d)] eta(d)=1/[1-theta(d)]。在整篇论文中，我们评论了旋转玻璃的异同。
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.