Dynamical Asymptotics Yield Gaussian Multiplicative Chaos for Hermitian Processes

The behaviour of random systems, such as the movement of particles, often reveals underlying mathematical patterns, and recent work explores these connections through the lens of probability and complex analysis. Ahmet Keles from the Courant Institute, New York University, along with collaborators, investigates how seemingly chaotic processes can converge to predictable distributions, building upon earlier findings in random matrix theory. The team demonstrates that systems governed by the laws of Brownian motion, when considered dynamically, exhibit a remarkable convergence to what is known as Gaussian multiplicative chaos, a complex mathematical object with applications in diverse fields. This research establishes a second link between random matrix theory and Liouville measures, extending previous results and providing new insights into the statistical properties of these systems, including the maximum of the characteristic polynomial and its rigidity.

This work rigorously demonstrates how fluctuations within certain random matrix models converge to these Gaussian fields, providing a mathematical foundation for observations previously made in diverse areas of physics and mathematics. The research extends previous studies of fluctuation fields and log-characteristic polynomials, confirming their universality across a broader range of random matrix ensembles and particle systems.

Random Matrices, Probability, and Limit Theorems

A comprehensive collection of references details the foundations of random matrix theory, probability, and related areas of mathematical physics. This bibliography covers a broad spectrum of topics, including eigenvalue distributions, characteristic polynomials, and universality results within various random matrix ensembles, such as the Gaussian Unitary and Orthogonal Ensembles. A strong emphasis on probability theory is evident, with significant coverage of stochastic processes, Gaussian processes, multiplicative chaos, and essential limit theorems like the Central Limit Theorem and the Law of Large Numbers. The list also highlights connections to integrable systems, Dyson processes, and applications in statistical mechanics.

References to multiplicative chaos, a random field arising in the study of random matrix characteristic polynomials, are particularly prominent, showcasing the work of researchers like Webb, Rhodes, and Vargas. Further areas of focus include the Ginibre ensemble, the Tracy-Widom distribution, and key contributions from authors like Pastur and Shcherbina. The bibliography also reflects interest in numerical methods, such as quadrature techniques, suggesting a practical approach to calculating quantities within random matrix theory. This resource serves as a valuable starting point for research, a comprehensive literature review, or the foundation for a graduate-level course. It also facilitates the identification of key researchers and potential gaps in current knowledge. The inclusion of recent publications from 2023 and 2024 indicates that this is a current and thorough collection of references, reflecting a deep understanding of the field and a particular focus on multiplicative chaos.

Gaussian Fields and Random Matrix Universality

Recent research has established a strong link between random matrix theory and the behaviour of log-correlated Gaussian fields, enhancing our understanding of complex systems exhibiting random fluctuations. This work rigorously demonstrates how fluctuations in specific random matrix models converge to these Gaussian fields, providing a mathematical basis for observations made in diverse areas of physics and mathematics. The research expands upon previous studies of fluctuation fields and log-characteristic polynomials, confirming their universality across a wider range of random matrix ensembles and particle systems. A key finding is the demonstration of how these fluctuations give rise to Gaussian multiplicative chaos (GMC) measures, complex probabilistic objects that describe the scaling limits of certain random processes.

The research provides a rigorous construction of these GMC measures, formally defined by an exponential weighting of a log-correlated Gaussian field, and establishes the conditions for their validity. This is significant because GMC measures appear as universal scaling limits in various models, offering a common framework for understanding seemingly disparate phenomena. Furthermore, the study extends these results to a dynamic setting, proving that these connections hold not just for static systems, but also for evolving processes described by non-intersecting Brownian motions. This dynamic generalization is achieved through the development of Fisher-Hartwig asymptotics, a mathematical tool for analysing the large-scale behaviour of these systems. The results show that the fluctuations exhibit optimal rigidity, meaning their behaviour is remarkably predictable despite the underlying randomness. This research builds upon and extends earlier work, providing a more complete and nuanced understanding of the interplay between random matrix theory and the fascinating world of log-correlated Gaussian fields and Gaussian multiplicative chaos.

Dynamical Asymptotics and Gaussian Multiplicative Chaos

The research establishes dynamical Fisher-Hartwig asymptotics, extending previous static results and demonstrating how these asymptotics apply to the Hermitian Ornstein-Uhlenbeck process. This work shows that fractional powers of the absolute value of the characteristic polynomial converge to a specific Gaussian multiplicative chaos measure, and provides a leading-order description of the log-characteristic polynomial alongside optimal bulk rigidity for non-intersecting Brownian motions. These findings represent a second connection between random matrix theory and Liouville measures, building upon earlier work and generalizing single-time convergence results. Furthermore, the study demonstrates a dynamic extension of existing results concerning the maximum of the log-characteristic polynomial and optimal rigidity properties.

The authors prove that, under the described dynamics, eigenvalues remain near their typical values with overwhelming probability over time windows proportional to the logarithm of the system size. This suggests a remarkable level of predictability in the behaviour of these complex systems, even as they evolve over time. The authors acknowledge that their results rely on certain approximations and assumptions within the mathematical framework, and that the complexity of the dynamics introduces limitations to the scope of their conclusions. Future research directions include exploring the implications of these findings for other random matrix ensembles and investigating the potential for extending these dynamical results to more general settings.