2024 Simons Collaboration on the Localization of Waves Annual Meeting

The fifth annual meeting of the Localization of Waves Collaboration sponsored by the Simons Foundation hosted 102 world-class scientists (95 in person and 7 remote) in New York City to discuss their work in understanding and exploiting the localization of waves brought about by a disordered environment or complex geometry, and related wave behaviors.

Collaboration Director, Svitlana Mayboroda (University of Minnesota/ETH), recipient of 2023 National Blavatnik award, kicked off the presentations with a talk entitled “Brownian Travelers and Boundary Complexity,” in which she presented groundbreaking results on the Robin harmonic measure on fractal and complex boundaries. Professor Bart van Tiggelen (Université Grenoble Alpes) followed with a talk on “The Theory of White Paint Revisited,” about the recent developments on the understanding of light propagation and localization in disordered media. Collaboration PI and 2022 Fields Medal recipient Hugo Duminil-Copin (University of Geneva/IHES) then discussed promising perspectives on mean-field approaches in high-dimensional systems in his talk, “A New Way of Looking at High-Dimensional Lattice Models.” Founding PI Douglas Arnold (University of Minnesota) then gave a talk with the provocative title, “What the @#$! is Cohomology Doing in Numerical Analysis?!” in which he detailed an elegant and powerful framework for solving numerically partial differential equations. Concluding the speakers on the first day was Professor David Huse (Princeton University) with a speech on “Many-Body Localization,” in which he presented the latest theoretical advances in the domain. Day one concluded with PI and speaker dinner at restaurant Portale.

Day two began with a talk by Horng-Tzer Yau (Harvard University) on “Random Band Matrices, Localization and Quantum Unique Ergodicity,” in which he discussed the mathematical progresses in understanding the nature of eigenvector localization in random band matrices versus usual Schrödinger equation. Following was founding PI Marcel Filoche (ESPCI), who presented in a talk entitled “The Structure of Anderson Transition,” a new understanding on the mobility edge of Anderson localization based on the localization landscape theory. The conference concluded with a stimulating presentation by Maseeh Professor of Mathematics Jeffrey Ovall (Portland State University), “Computational Tools for Exploring Eigenvector Localization.” Professor Ovall presented new methods to detect the presence of localized eigenfunctions and to predict their shape, in particular in the context of magnetic Schrödinger operators.

A gratifying aspect of the meeting was that, despite the wide diversity of backgrounds of the participants, there were lively discussions across disciplinary boundaries throughout.

The Annual meeting allowed for the collaboration to again gather and discuss our goals and objectives as we continue our wide reaching and highly impactful studies.

• Svitlana Mayboroda and Eugenia Malinnikova will pursue the landscape approach to the structure of Wigner functions in disordered systems.
• Guy David, Svitlana Mayboroda, Doug Arnold and Marcel Filoche will develop new approaches to describe the behavior of the solution to the Laplace equation with Robin (or Fourier) boundary conditions next to complex interfaces.
• James Speck, together with Claude Weisbuch, will explore the role of impurity disorder combined with alloy disorder in the performances of LEDs.
• Marcel Filoche, together with Claude Weisbuch and James Speck, will develop Wigner-Weyl-based computations of radiative and non-radiative recombination in disordered semiconductor compounds.
• Alain Aspect is going to develop cutting edge experiments for controlling interactions in disordered cold atom systems.
• Marcel Filoche and Svitlana Mayboroda will explore connections between the percolation phenomena for the landscape and the mobility edge in tight-binding models.

We are grateful to the Simons Foundation for the opportunity to continue our research into this exciting topic and look forward to sharing even more results of our efforts at the next Simons Foundation annual meeting, February 20 and 21, 2025.

Svitlana Mayboroda
University of Minnesota and ETH Zurich

Brownian Travelers & Boundary Complexity

Svitlana Mayboroda will discuss how the Brownian travelers see irregular, disordered, possibly higher dimensional boundaries, and how geometric complexity, in nature or in engineering, enhances robustness and efficiency of the underlying physical systems. From the point of view of mathematics, these are notoriously difficult problems of the structure and support of the harmonic measure, as well as regularity of the emerging free boundaries. Applications in physics range from the construction of noise abatement walls to transport in lung to heterogeneous catalysis. In particular, we prove that contrary to the conjectures put forward by physicists and contrary to the “classical” Dirichlet scenario, the harmonic measure associated to a partially reflective Brownian motion is absolutely continuous with respect to the Hausdorff measure on any boundary.
 

Bart van Tiggelen
Université Grenoble Alpes

The Theory of White Paint Revisited
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Numerical simulations have demonstrated the absence of an Anderson transition when light propagates in a dense 3D medium filled with electric dipoles. They have no easy explanation, and all existing theories predict a mobility edge when the Ioffe-Regel parameter (a product of wavenumber and mean free path) is of the order of one. Some crucial element must have been overlooked. Longitudinal electric waves are well known in classical and quantum electromagnetism, but their role has so far been underestimated in radiative transfer. The reason is that, alone, they cannot propagate because they do not create a magnetic field and therefore no Poynting vector, as is the case for transversely polarized waves. We have reconsidered the transport theory of electromagnetic waves in 3D random media and found that the interference of longitudinal and transverse waves creates a second channel in transport. Because the two channels are coupled by scattering, it is much more difficult to close both simultaneously and to have Anderson localization.
 

Hugo Duminil-Copin
Université de Genève

A New Way of Looking at High-Dimensional Lattice Models

Embarking on an exploration of high-dimensional lattice models, this presentation delves into contemporary advancements within the realm of percolation, self-avoiding walks, the Ising model, and the XY model. Our approach diverges from traditional lace-expansion and renormalization techniques, opting for an innovative perspective that elucidates mean-field behavior. The crux of our methodology lies in an examination of the random-walk representation inherent in these intricate models.
 

Douglas Arnold
University of Minnesota

What the @#$! is Cohomology Doing in Numerical Analysis?!
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As the name suggests, numerical analysis — the study of computational algorithms to solve mathematical problems, such as systems of differential equations — has traditionally been viewed mostly as a branch of analysis. Geometry, topology and algebra played little role. However, in the last decade or so, things have changed. The recent literature on numerical analysis is replete with papers using concepts that are new to the subject, say, symplectic differential forms or de Rham cohomology or Hodge theory. In this talk, Douglas Arnold will discuss some examples of this phenomenon, especially the finite element exterior calculus. We shall see why these new ideas arise naturally in numerical analysis and how they contribute.
 

David Huse
Princeton University

Many-Body Localization

Many-body localization (MBL) is Anderson localization of many interacting quantum degrees of freedom in highly-excited states at conditions that correspond to a nonzero entropy density at thermal equilibrium. The opposite of MBL is thermalization, where the isolated quantum many-body system successfully acts as a thermal bath for itself, bringing all of its small subsystems to thermal equilibrium with each other via the unitary quantum dynamics of the closed system. MBL, unlike wave or single-particle localization, does not appear to have a weak localization regime.

For systems with short-range interactions, the transition from thermalization to MBL occurs in two stages as the interactions are reduced: First is a smooth crossover to a “glassy” prethermal MBL regime, where the thermalization time of a large system becomes extremely large but not infinite. Then, at still weaker interaction is the dynamical phase transition in to the MBL phase, which in some cases occurs at a strength of interactions that is so small that it is thermodynamically insignificant in the limit of large systems, even though it has strong long-time dynamical effects.
 

Horng-Tzer Yau
Harvard University

Random Band Matrices, Localization and Quantum Unique Ergodicity

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Marcel Filoche
ESPCI Paris

The Structure of the Anderson Transition
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The structure of the phase diagram of eigenfunction localization for tight-binding Hamiltonians with random independently and identically distributed (i.i.d.)P disorder (à la Anderson) in the energy-disorder plane is well known: below dimension d=2, all eigenfunctions are localized for non-vanishing disorder, and above d=2, a delocalized phase appears separated from the localized phase by a transition line called the “mobility edge,” predicted by the so-called self-consistent theory of localization in the case of uniform disorder. Marcel Filoche will show that behind this simple description, there is in fact a more complicated structure emerging already at a low dimension and will explore the various mechanisms at work in the localization/delocalization transition.
 

Jeffrey Ovall
Portland State University

Computational Tools for Exploring Eigenvector Localization
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We present an algorithm for determining all eigenpairs of the magnetic Schrödinger operator whose eigenvalues lie within a user-specified range, and whose eigenvectors are sufficiently concentrated within a user-specified subdomain (not just “ground states”) or certifying that no such eigenpairs exist. The algorithm is based on a relatively compact perturbation of the magnetic Schrödinger operator that is designed to highlight only strong candidates for eigenpairs satisfying the given criteria. Heuristic, theoretical and computational support will be provided for the algorithm. Although the theoretical results apply to relatively generic magnetic Schrödinger operators, the numerical illustrations will focus on the two “extreme” cases: the standard Schrödinger operator and the magnetic Laplacian operator. In contrast to the standard Schrödinger operator, for which a rich mathematical theory based on the localization landscape provides a fairly detailed understanding of localization in the lower part of the spectrum, not much is known about the mechanisms driving localization for the magnetic Laplacian, and we provide some theoretical and computational insight.