High-dimensional probability in ergodic theory and statistical physics

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

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.