Quenched Tails and Almost Sure Limit Laws

评估罕见事件的概率有助于理解它们发生的机制，当许多不同的罕见事件可能发生时，它确定导致其中一个事件实际发生的机制。这种方法非常成功，通常是在两个随机性水平存在时找到熄灭的渐近行为，条件是一个随机性元素的固定的、典型的实现。这项工作主要集中在两个方向上的应用：1.实验和模拟涉及的动力系统失去平衡报告了老化现象，即“记忆”积累在系统中。统计物理中的涨落耗散定理将平衡态关联函数的时间导数与动力学方程中微小微扰的影响联系起来。物理学文献中预测，衰老与涨落耗散定理的一种特定形式的解体有关。拟议活动的一个目标是形成涨落-耗散定理的数学版本，了解衰老机制的原因，并将两者严格联系起来。高斯自由场和布朗运动是两个基本的随机物体，具有极大的内在美，对概率论(以及数学物理)也很感兴趣。PI和他的合作者已经非常成功地研究了与平面布朗运动和随机游动相关的占领测度和覆盖过程的极值点的分形几何。PI的学生、他们的合作者和其他人最近的工作表明，高斯自由场的极值与布朗运动的极值之间有很强的相似性，以及随机高度函数的轮廓线、高斯自由场的轮廓线和某些共形不变环系综之间的新关系。进一步追求这种相似性，提出的活动的一个目标是获得新的见解，并在平面高斯自由场和布朗运动之间建立新的联系。在过去的三十年里，物理界开发了复杂的非严格技术，用于对随机动力学、吉布斯测量和具有共形对称性的平面对象的渐近性进行非常准确的预测。最近，数学家们开始在发展相应的严格理论和证明其中一些预测方面取得重大进展。这一过程中的关键工具之一是大偏差理论及其背后的直觉。拟议的活动部分是由这种不严格的物理预测推动的。因此，预计它将产生新的数学思想和技术，除了立即解决概率学家非常感兴趣的挑战性问题外，还将对概率论和统计物理之间的接口产生影响。这笔助学金将为学生提供概率论方面的研究生培训。这项提案中将要开发的技术还有望在未来应用于普遍非线性过滤的理论和实践，这是一个与传播理论相关的问题(有关一些相关活动和预印本，请参阅网址www.stat.stanford.edu/~amir/)。

0406042Dembo The evaluation of probabilities of rare events leads to an understanding of the mechanisms by which they can occur, and when many different rare events are possible, it identifies the mechanism causing one of them to actually occur. This approach has been very successful, often for finding the quenched asymptotic behavior when two levels of randomness are present, conditioned on a fixed, typical realization of one element of randomness. The proposed activity focuses on the applications of this approach in two directions:1. Experiments and simulations involving dynamical systems out of equilibrium report an aging phenomenon, that is "memory" being accumulated in the system. The fluctuation-dissipation theorem of statistical physics relates the time derivative of the correlation function at equilibrium to the effect of small perturbations in the dynamical equations. It is predicted in the physics literature that aging is related to a specific form of breakup of the fluctuation-dissipation theorem. One goal of the proposed activity is to form the mathematical version of the fluctuation-dissipation theorem, to understand the causes of the aging mechanism, and to rigorously relate the two.2. The Gaussian free field and the Brownian motion are two fundamental random objects of great intrinsic beauty and much interest in probability theory (as well as in mathematical physics). The PI and his collaborators have been very successful in studying the fractal geometry of the extremal points of the occupation measure and covering process associated with planar Brownian motion and random walks. Recent works by the PI's students, their collaborators, and others, suggest strong similarity between the extremes of the Gaussian free field and those of the Brownian motion, as well as novel relations between contour lines of random height functions, those of the Gaussian free field and certain conformally invariant loop ensembles. Pursuing further this similarity, one goal of the proposed activity is to gain new insights and develop novel connections between the planar Gaussian free field and Brownian motion. Over the past three decades, the physics community has developed sophisticated non-rigorous techniques for making very accurate predictions about the asymptotics of random dynamics, Gibbs measures, and planar objects with conformal symmetries. More recently, mathematicians started making significant progress in developing the corresponding rigorous theories and proving some of these predictions. One of the key tools in this process has been the theory of large deviations and the intuition behind it. The proposed activity is partially motivated by such non-rigorous physics predictions. As such, it is expected to result in new mathematical ideas and techniques that, beyond the immediate resolution of challenging problems of much interest to probabilists, would have an impact on the interface between probability theory and statistical physics. This grant will provide graduate training of students in probability theory. The techniques to be developed in this proposal are also expected to have future applications to the theory and practice of universal non-linear filtering, a problem of relevance and much interest for communication theory (see the URL www-stat.stanford.edu/~amir/ for some related activity and preprints).