Mean field asymptotic for stochastic processes on graphs

在过去的三十年里，物理学家已经开发出精密的非严格技术来精确预测大型复杂（随机）系统的渐近行为。数学家们在发展相应的严格理论和证明这些预测方面取得了重大进展。概率论是这种融合的前沿，从大偏差理论开始，继续在相互作用的粒子、大随机矩阵、吉布斯测量和保形对称的平面物体的随机动力学研究中出现的活跃活动。从物理学的角度来看，一个特别成功的例子是平均场（无序）模型，其中通常存在两个层次的随机性，其中一个是冻结的（淬灭的），以依赖性图或权重的形式出现，其分布在无序上平均时是可交换的。我们计划追求的具体方向是：(1)严格研究根据（随机选择的）图集合确定的平均场模型强相互作用的大型离散变量系统，(2)随机矩阵结构模型谱的渐近行为与相应随机图和图嵌入的性质之间的新联系，(3)非平衡自旋系统的随机动力学性质，例如时效和有限尺寸的缩放与它们的相变有关。该项目有望产生新的数学思想和技术，以解决概率论中的挑战性问题，影响其与统计物理的接口，并为大型约束满足网络的典型行为分析提供见解，从而推动NSF网络支持发现和创新计划的核心进展。对研究生和研究生培养离散概率和信息论的学生也有长期的影响。

Over the past three decades, physicists have developed sophisticated non-rigorous techniques for accurately predicting the asymptotic behavior of large complex(random) systems. Mathematicians are making significant progress in developing the corresponding rigorous theories and proving 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. A particular success story from a physics point of view are mean field(disordered) models, where two levels of randomness are often present, one of whom frozen (quenched), in the form of dependence graph or weights for the resulting stochastic process, whose distribution is exchangeable when averaged over the disorder. Specific directions we plan to pursue are (1) Rigorous study of large systems of discrete variables that are strongly interacting according to a mean field model determined by an ensemble of (randomly chosen) graphs, (2) 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,(3) properties of stochastic dynamics for spin systems out of equilibrium, such as aging and the finite size scaling associated with their phase transitions.This project is expected to yield new mathematical ideas and techniques that resolve challenging problems in probability theory, impact its interface with statistical physics, and provide insight to the analysis of typical behavior of large constraint satisfaction networks, thus advancing progress at the core of the NSF cyber-enabled discovery and innovation initiative. Long-term effects on graduate and post-graduate training of students in discrete probability and information theory are also expected.