High-dimensional probability in ergodic theory and statistical physics

在数学概率论中，平稳随机过程模拟的现象是，随着时间的推移，出现新的不可预测的结果，但其潜在的概率定律保持不变：最简单地考虑重复抛硬币的情况。遍历理论抽象地研究在这样一个过程中可能存在什么样的随机性，其中两个过程被认为是等价的，如果其中一个过程可以通过对另一个过程进行适当的编码来获得。然而，在现代编码理论、统计学和数据科学中，许多非常重要的随机现象并不是以平稳过程中由时间箭头指示的线性方式出现的。相反，它们是根据一个庞大的基础网络构建的，而这个网络本身可能是高度混乱的。当前的项目寻求使遍历理论的方法适应这种新的环境，或者理解它们失败的原因，然后开发替代方法。对于由底层无序网络控制的随机性模型，需要研究的一些主要问题包括：(I)广泛的统计特征是否以简单的方式随网络的大小而缩放；(Ii)在网络的特定节点处可见的随机行为是否可以从模型的参数中容易地估计；(Iii)如何有效地确定给定的网络随机性模型是否可以以某种更简单的形式重新编码，例如每个节点处的纯独立噪声。所有这些问题都出现在计算机和数据科学的现代研究的理论基础上，并可能成为这些领域新发现的指南。对这些问题的追求需要概率和纯数学的不同部分的背景知识，因此目前的项目包括开发教材和培训研究生学习这些学科的新颖但有价值的组合的计划。该项目还有两个更具体的目标。第一个是首席研究者最近证明遍历理论中的弱Pinsker定理的方法的进一步发展。这个定理断言，在测度论同构的意义上，任何平稳的遍历过程都可以被重新编码为几乎确定的分量和纯随机分量。证明中的方法适用于平稳随机变量序列的联合分布，从联合分布的一些基本信息论参数导出结构描述。这种影响仍处于初级阶段，拟议的项目将探讨一系列相关参数及其后果。第二个目的是关于组合学和统计物理中广泛而重要的联合分布，它们是在局部树形图上构造的。这些模型提供了自由基团保持测量的行为的例子，反过来，自由基团遍历理论的方法有助于描述它们的渐近行为。物理学文献预测了新的和相应的现象，如破碎和凝聚，但只有几个特殊情况被严格地建立起来。该项目将使用遍历理论中的概念和方法来探索这些现象。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

In mathematical probability theory, stationary stochastic processes model phenomena that reveal new, unpredictable outcomes as time passes, but whose underlying probability laws remain constant: consider, most simply, the repeated tossing of a coin. Ergodic theory studies abstractly what kinds of randomness are possible in such a process, where two processes are regarded as equivalent if either can be obtained by a suitable encoding of the other. However, many random phenomena of great importance in modern coding theory, statistics, and data science do not arise in the linear fashion that is dictated by the arrow of time in a stationary process. Rather, they are structured according to a large underlying network, which may itself be highly disordered. The current project seeks to adapt methods of ergodic theory to this new setting or understand why they fail and then develop replacements. For models of randomness governed by underlying disordered networks, some of the main issues to be investigated include: (i) whether the broad statistical features scale in a simple way with the size of the network; (ii) whether the random behavior that is visible at a particular node of the network can be estimated easily from the parameters of the model; (iii) how one can determine effectively whether a given model of network randomness can be re-encoded in some simpler form, such as pure independent noise at each node. All of these issues arise in the theoretical underpinnings of modern research in computer and data science and could serve as a guide towards new discoveries in those fields. The pursuit of these questions requires background knowledge from diverse parts of probability and pure mathematics, and so the current project includes plans to develop educational materials and train graduate students in a novel but valuable mix of those disciplines.The project has two more specific aims. The first is the further development of the methods in the principal investigator's recent proof of the weak Pinsker theorem in ergodic theory. This theorem asserts that any stationary ergodic process may be re-encoded, in the sense of measure-theoretic isomorphism, into a nearly deterministic component and a purely random component. The methods from the proof apply to the joint distribution of a stationary sequence of random variables, deriving a structural description from some basic information-theoretic parameters of a joint distribution. Such implications are still in their infancy, and the proposed project would explore a range of related parameters and their consequences. The second aim concerns the broad and important class of joint distributions in combinatorics and statistical physics that are constructed over locally tree-like graphs. These models provide examples of measure-preserving actions of free groups, and conversely methods from free-group ergodic theory help to describe their asymptotic behavior. The physics literature predicts novel and consequential phenomena such as shattering and condensation, but only a few special cases have been established rigorously. The project will explore these phenomena using notions and methods from ergodic theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.