Polynomial chaos and scaling limits of disordered systems
Polynomial chaos and scaling limits of disordered systems
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DOI：
10.4171/jems/660
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发表时间：
2013-12
影响因子：
2.6
通讯作者：
F. Caravenna;Rongfeng Sun;Nikos Zygouras
中科院分区：
文献类型：
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作者：
F. Caravenna;Rongfeng Sun;Nikos Zygouras
受 Alberts、Khanin 和 Quastel 最近工作的启发，我们制定了一般条件，确保独立随机变量的多线性多项式序列（称为多项式混沌展开）收敛到由 d 维白噪声上的维纳混沌展开给出的极限随机变量。我们的方法的一个关键要素是多项式混沌展开的 Lindeberg 原理，它扩展了 Mossel、O'Donnell 和 Oleszkiewicz 的早期工作。这些结果提供了一个统一的框架来研究与无序相关的统计力学系统的连续和弱无序标度极限，包括无序钉扎模型、1+1 维（长程）定向聚合物模型和二维随机场 Ising 模型。这为疾病相关性的研究提供了新的视角，并产生了值得进一步研究的有趣的新连续体模型。
Inspired by recent work of Alberts, Khanin and Quastel, we formulate general conditions ensuring that a sequence of multi-linear polynomials of independent random variables (called polynomial chaos expansions) converges to a limiting random variable, given by a Wiener chaos expansion over the d-dimensional white noise. A key ingredient in our approach is a Lindeberg principle for polynomial chaos expansions, which extends earlier work of Mossel, O'Donnell and Oleszkiewicz. These results provide a unified framework to study the continuum and weak disorder scaling limits of statistical mechanics systems that are disorder relevant, including the disordered pinning model, the (long-range) directed polymer model in dimension 1+1, and the two-dimensional random field Ising model. This gives a new perspective in the study of disorder relevance, and leads to interesting new continuum models that warrant further studies.