Quenched Tails and Almost Sure Limit Laws

0406042Dembo对罕见事件概率的评估可以让我们理解它们发生的机制，当许多不同的罕见事件都有可能发生时，它可以识别出导致其中一个事件实际发生的机制。这种方法非常成功，通常用于发现存在两级随机性时的淬灭渐近行为，条件是一个随机元素的固定的、典型的实现。拟议的活动侧重于这一办法在两个方面的应用：涉及非平衡动力系统的实验和模拟报告了一种老化现象，即“记忆”在系统中积累。统计物理的涨落耗散定理将平衡状态下相关函数的时间导数与动力学方程中小扰动的影响联系起来。物理文献预测，老化与涨落耗散定理的一种特定破坏形式有关。所提出的活动的一个目标是形成波动-耗散定理的数学版本，了解老化机制的原因，并严格地将两者联系起来。高斯自由场和布朗运动是两个基本的随机物体，具有内在的美，在概率论（以及数学物理）中非常有趣。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行走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，高斯自由场的极端和布朗运动的极端之间有很强的相似性，以及随机高度函数的等高线，高斯自由场的等高线和某些保形不变环系之间的新关系。进一步追求这种相似性，所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，物理界已经开发出了复杂的非严格技术，用于非常准确地预测随机动力学、吉布斯测量和具有保形对称的平面物体的渐近性。最近，数学家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。这个过程中的关键工具之一是大偏差理论及其背后的直觉。这种提议的活动部分是由这种非严格的物理预测驱动的。因此，它有望产生新的数学思想和技术，除了直接解决概率学家感兴趣的具有挑战性的问题外，还将对概率论和统计物理学之间的接口产生影响。本基金将为研究生提供概率论培训。本提案中开发的技术也有望在通用非线性滤波的理论和实践中有未来的应用，这是一个与通信理论相关且非常感兴趣的问题（参见URL www-stat.stanford.edu/~amir/获取一些相关活动和预印本）。

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).