Mean Field Asymptotic for Stochastic Processes on Graphs

在这个建议的共同点是集中在平均场无序模型最初在统计物理学。这些通常以两个随机性水平为特征，其中一个被冻结（淬火），以所产生的随机过程的依赖图或权重的形式，其分布是可交换的（因此是平均场模型）。 关于这些模型，我们关注以下方向。对离散变量大系统的概率模型的严格研究，这些变量根据随机图结构强烈相互作用。 其目的是发展一个更好的数学理解他们的相变和存在的多个吉布斯措施。一个相关的努力与随机矩阵的结构模型的谱的渐近行为和相应的随机图和图嵌入的属性之间的新的连接的发展。理解自旋系统的一些意外的渐近行为背后的范围和原因平衡。一个突出的例子是老化现象，其中记忆在系统中随着时间的推移而积累，以及它与波动耗散定理（将平衡时相关函数的时间导数与马尔可夫动力学中小扰动的影响联系起来）的一种尚未理解的特定形式的破裂的关系。在过去的三十年里，物理学家已经开发出复杂的非严格技术，用于精确预测大型复杂随机系统的渐近行为。他们的研究动机是希望了解物质的各种状态的集体行为，以及连接它们的相变。例如，理解水在临界温度下变成蒸汽的目标，或者理解磁体在临界温度下失去自发磁化的目标，激发了临界现象的复杂统一理论的发展。随着时间的推移，物理学家开发的模型和直觉显然依赖于深刻的数学原理，它的影响范围比简单的物理系统要广得多。最近，数学家们在发展相应的严格理论和证明其中一些预言方面取得了重大进展。概率论是在这一收敛的前沿，从大偏差理论开始，并继续在相互作用的粒子，大随机矩阵，吉布斯措施和平面物体与共形对称的随机动力学研究中出现的充满活力的活动。这个项目的重点是发展这一研究线，特别是发展没有（有限维）几何结构的大型随机系统的数学。这些被物理学家称为“平均场系统”，并且具有显着的普遍性。

The common thread in this proposal is the focus on mean field disordered models originally developed within statistical physics. These are usually characterized by two levels of randomness, one of whom frozen (quenched), in the form of dependence graph or weights for the resulting stochastic process, whose distribution is exchangeable (hence being a mean field model). In connection with such models we focus on the following directions. The rigorous study of probabilistic models for large systems of discrete variables that are strongly interacting according to a random graph structure. The aim is to develop a better mathematical understanding of their phase transitions and the existence of multiple Gibbs measures. A related effort has to do with the development of novel connections between the asymptotic behavior of the spectrum of structured models of random matrices and properties of the corresponding random graphs and graph embeddings.Understanding the scope and reasons behind some unexpected asymptotic behavior of stochastic dynamics for spin systems out of equilibrium. A prominent example is the aging phenomenon where memory accumulates in the system with time and its relation to a yet to be understood specific form of breakup of the fluctuation-dissipation theorem (which relates the time derivative of the correlation function at equilibrium to the effect of small perturbations in Markovian dynamics). Over the past three decades, physicists have developed sophisticated non-rigorous techniques for accurately predicting the asymptotic behavior of large complex random systems. Their studies were motivated by the desire to understand the collective behavior of various states of the matter, and the phase transitions connecting them. For instance, the objective of understanding water at the critical temperature at which it turns into vapor, or magnets at the critical temperature at which they lose spontaneous magnetization, has motivated the development of a sophisticated unified theory of critical phenomena.Over time, it has become apparent that the models and intuitions developed by physicists rest on deep mathematical principles, whose reach is much broader than simply physical systems. More recently, mathematicians are making significant progress in developing the corresponding rigorous theories and proving some of these predictions. Probability theory is at the forefront of this convergence, starting with the theory of large deviations and continuing with the emerging vibrant activity in the study of stochastic dynamics of interacting particles, large random matrices, Gibbs measures and planar objects with conformal symmetries.This project focuses on developing this line of research, in particular to develop the mathematics of large random systems without (finite-dimensional) geometric structures. These are called by physicists `mean field systems' and enjoy a remarkable degree of universality.