The Universality Classes in the Parabolic Anderson Model
The Universality Classes in the Parabolic Anderson Model
复制标题
抛物线安德森模型中的普遍性类
DOI：
10.1007/s00220-006-0075-4
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发表时间：
2005
影响因子：
2.4
通讯作者：
Peter Mörters
中科院分区：
文献类型：
--
作者：
R. Hofstad;W. König;Peter Mörters
本文讨论了抛物型安德森模型的长时间行为，以及具有随机位势的热方程在\mathbb{Z}^{d}上的柯西问题.我们认为一般i.i.d.的潜力，并表明，正是四种不同类型的间歇性行为可以发生。这四个普适性类依赖于势分布的上尾：（1）比双指数尾厚的∞处的尾，（2）由Gärtner和Molchanov研究的∞处的双指数尾，（3）一个称为几乎有界势的新类，以及（4）由Biskup和König研究的从上面有界的势。新的类（3），其中包含无界和有界的潜力，研究在退火和淬火设置。我们发现，无边界增加的岛屿，其直径是缓慢变化的时间上发生的波动。描述的最佳配置文件的潜力和解决方案的特征变分公式明确解决抛物线，分别高斯密度。我们对第（3）类的分析依赖于连续时间简单随机游动局部时的两个大偏差结果。其中一个结果已由Brydges和前两位作者在[BHK 04]中证明，并在此用来修正[BK 01]中的一个证明。
We discuss the long time behaviour of the parabolic Anderson model, the Cauchy problem for the heat equation with random potential on$$\mathbb{Z}^{d}$$. We consider general i.i.d. potentials and show that exactly four qualitatively different types of intermittent behaviour can occur. These four universality classes depend on the upper tail of the potential distribution: (1) tails at ∞ that are thicker than the double-exponential tails, (2) double-exponential tails at ∞ studied by Gärtner and Molchanov, (3) a new class called almost bounded potentials, and (4) potentials bounded from above studied by Biskup and König. The new class (3), which contains both unbounded and bounded potentials, is studied in both the annealed and the quenched setting. We show that intermittency occurs on unboundedly increasing islands whose diameter is slowly varying in time. The characteristic variational formulas describing the optimal profiles of the potential and of the solution are solved explicitly by parabolas, respectively, Gaussian densities. Our analysis of class (3) relies on two large deviation results for the local times of continuous-time simple random walk. One of these results is proved by Brydges and the first two authors in [BHK04], and is also used here to correct a proof in [BK01].