Mean Field Asymptotic for Stochastic Processes on Graphs

这一提议的共同主线是关注最初在统计物理学中发展起来的平均场无序模型。这些通常具有两个随机性级别的特征，其中一个以依赖图或所产生的随机过程的权重的形式冻结(熄灭)，其分布是可交换的(因此是平均场模型)。关于这类模型，我们主要关注以下几个方向。对离散变量大系统的概率模型的严格研究，这些离散变量系统按照随机图结构强相互作用。这样做的目的是为了更好地从数学上理解它们的相变和多重吉布斯度量的存在。一个相关的工作是在随机矩阵的结构模型的谱的渐近行为与相应的随机图和图嵌入的性质之间建立新的联系。理解自旋系统失去平衡的随机动力学的一些意外的渐近行为的范围和原因。一个突出的例子是老化现象，即记忆随着时间在系统中积累，以及它与涨落-耗散定理(将平衡点的关联函数的时间导数与马尔可夫动力学中的小扰动的影响联系在一起)的特定形式的破裂的关系尚不清楚。在过去的三十年里，物理学家已经开发出复杂的、非严谨的技术来准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水蒸气的目标，或在水失去自发磁化的临界温度下理解磁体的目标，推动了关于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建立在深刻的数学原理之上，这些原理的覆盖范围比简单的物理系统广泛得多。最近，数学家们在发展相应的严格理论和证明其中一些预测方面取得了重大进展。概率论处于这种收敛的前沿，它从大偏差理论开始，继续研究相互作用的粒子、大随机矩阵、Gibbs测度和具有共形对称的平面对象的随机动力学的新的振动活动。本项目致力于发展这一研究方向，特别是发展没有(有限维)几何结构的大型随机系统的数学。它们被物理学家称为“Mean场系统”，具有惊人的普适性。

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.