Schedule for: 24w5237 - Randomness and Quasiperiodicity in Mathematical Physics

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Informal Gathering at the BIRS Lounge in PDC (2nd Floor)

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

The spectrum of a discrete Schrodinger operator with periodic potential is known to be a finite union of intervals. The same is true for the Anderson Model, i.e. for a Schrodinger operator where the potential is defined be a sequence of iid random variables. The “intermediate” case of deterministic aperiodic potentials, or “one dimensional quasicrystals” (Fibonacci Hamiltonian, Sturmian, Almost Mathieu, limit periodic, substitution potentials, etc.), tend to present a Cantor set as a spectrum, even if it is not easy (or even notoriously hard) to prove in many cases. What happens if one adds some random noise on top of an aperiodic potential, or, more generally, a given ergodic potential? It turns out that in many cases “randomness” wins, both in terms of spectral type and in terms of the topological structure of the spectrum. More specifically, one can prove Anderson Localization for such models, i.e. to show that the spectrum must be pure point almost surely. And, under the additional assumption that the phase space of the dynamical systems that defines the background potential is connected, one can show that the almost sure spectrum must be a finite union of intervals, exactly as in the Anderson Model. In particular, the spectrum of a quasiperiodic 1D Schrodinger operator with iid random noise is a finite union of intervals. The talk is based on a joint project with V.Kleptsyn, as well as on recent results joint with A.Avila and D.Damanik.

In this talk, I will introduce some recent joint work with A. Avila and D. Damanik in showing Anderson localization for Schrodinger operators generated by hyperbolic transformations. Specifically, we consider a topological mixing subshift of finite type with an ergodic measure admitting a bounded distortion property. We show that if the Lyapunov exponent has uniform positivity and uniform LDT on a compact interval, then the operator has Anderson localization on that interval almost surely. For potentials which have small supremum norms or are locally constant, we established a certain uniform LDT which together with our previous work on positivity of the LE yields full spectral localization. In particular, our work can be applied to the doubling map, the Arnold's cat map, or the Markov chain.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We consider the simple random walk on Galton–Watson trees with supercritical offspring distribution, conditioned on non-extinction. In case the offspring distribution has finite support, we prove an upper bound for the annealed return probability to the root which decays subexponentially in time with exponent 1/3. This exponent is optimal. Our result improves the previously known subexponential upper bound with exponent 1/5 by Piau [Ann. Probab. 26, 1016–1040 (1998)]. For offspring distributions with unbounded support but sufficiently fast decay, our method also yields improved subexponential upper bounds.

Topological insulators are usually studied using a spectral gap condition at the Fermi energy. However, physically it is more interesting to employ the insulator condition via Anderson localization, i.e., forgo a spectral gap and assume the Fermi energy is surrounded by eigenvalues corresponding to localized states. I will describe the problem of topological classification of insulators, in particular in one-dimension, in this Anderson localized regime of insulators.

Transport in disordered solids is a phenomenon involving many actors. The motion of a single quantum particle in such a solid is described by a random Hamiltonian. Transport involves many interacting particles, usually, a small fraction of the particles present in the material. One striking phenomenon observed and proved in disordered materials is localization: disorder can prevent transport! While this is quite well understood at the level of a single particle, it is much less clear what happens in the case of many interacting particles. Physicist proposed a number of tools (exponential decay of finite particle density matrices, entanglement entropy, etc) to discriminate between transport and localization. Unfortunately, these quantities are very difficult to control mathematically for "real life" models. We'll present a toy model where one can actually get a control on various of these quantities at least for the ground state of the system. The talk is based on the PhD theses of and joint work with N. Veniaminov and V. Ognov.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

No half-line Schrodinger operator or Jacobi matrix generated by the doubling map has any (essential) spectral gaps. The proof uses Johnson's formulation of gap-labelling: the value of the IDS in the gap coincides with the value of the Schwartzman homomorphism applied to the stable section of the transfer matrix cocycle. This enables one to extract a contradiction from the assumption of a nontrivial gap. [The talk contains joint work with D. Damanik, I. Emilsdottir, and Z. Zhang]

Non-self-adjoint operators have attracted widespread attention due to their importance in mathematical physics. In particular, the theory of the quasi-periodic Schr\"odinger operator with complex-valued potential has made many breakthroughs in recent years. I will report some recent progress and leave some open questions on this topic.

Many spectral properties of 1D Schr\"odinger operators have been linked to the Lyapunov exponent of the corresponding Schr\"odinger cocycle. While the situation for one-frequency quasi-periodic operators with analytic potential is well-understood, the multifrequency and non-analytic situation is not. The purpose of this talk is twofold: first, discuss our recent work on multi-frequency analytic quasi-periodic cocycles, establishing continuity (both in cocycle and jointly in cocycle and frequency) of the Lyapunov exponent for non-identically singular cocycles, and second, discuss ongoing work extending these results to suitable Gevrey classes. Analogous results for analytic one-frequency cocycles have been known for over a decade, but the multi-frequency results have been limited to either Diophantine frequencies (continuity in cocycle) or SL(2,C) cocycles (joint continuity). We will discuss the main points of our argument, which extends earlier work of Bourgain.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Topological insulators (TI) are a class of 2D materials which behave like insulators in their bulk but support robust states along their edges. One of the key property of TI that is expected to be true is the robustness of the property above w.r.t. the shape of the edge. In this talk, we will discuss how does shape of edge influence the property of TI above. In particular, we will both give a general, intuitive condition for this property to hold, and provide a counter-example otherwise. We also show why in practical situation, experiments may provide misleading results on TI. This work is based on a joint work with Alexis Drouot.

We will discuss the duality approach to Avila's global theory, developed in Ge-J-You-Zhou, that led to solutions of several outstanding spectral problems, including the almost reducibility conjecture (L. Ge) and the ten martini problem (Cantor spectrum with no condition on irrational frequencies), previously known only for the almost Mathieu operator, for a large class of one-frequency analytic quasiperiodic operators, including nonperturbative analytic neighborhoods of several popular explicit families (Ge-J-You). We will then give more detail on the latter proof.

In this talk, we proved for a class of almost periodic CMV matrices, there exist explicit mobility edges. That is, there is a coexistence of the absolutely continuous part and the pure point part of the spectrum. Moreover, the spectral measure restricted to the ac part of the spectrum is purely absolutely continuous. In the pure point part, we proved the Anderson localization with explicit exponential decay rate of the eigenfunctions. The proof is based on the technique of computing Lyapunov exponents through Avila's global theory and an observation of symmetries underlying the structure of generalized CMV matrices.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

It is often expected that the local statistical behavior of eigenvalues of some system depends only on its local properties; for instance, the local distribution of zeros of orthogonal polynomials should depend only on the local properties of the measure of orthogonality. The most commonly studied case is known as bulk universality, where Christoffel-Darboux kernels have a double scaling limit given by the sine kernel. In this talk, I will discuss a new approach which gave the first completely local sufficient condition for bulk universality, and a second new approach which gives necessary and sufficient conditions for universality limits. This work uses the de Branges theory of canonical systems, and applies to other self-adjoint systems with 2x2 transfer matrices such as Schrodinger operators. The talk is based on joint work with Benjamin Eichinger (TU Wien), Brian Simanek (Baylor University), and Harald Woracek (TU Wien).

We discuss the behavior of the NLS on R or the discrete NLS on Z with bounded initial data. Examples include quasi-periodic and random data. On Z, polynomial bounds are proved for all bounded initial data. In the continuum, local existence is established for real analytic data. We also discuss the long time behavior of a smoothly driven anharmonic oscillator. This is joint work with B. Dodson and A. Soffer.

I will review properties of two classes of moire materials, twisted bilayer graphene (TBG) and twisted semiconductors (TMDs) with an emphasis on their mathematical differences. If time permits, I will discuss results on these models under disorder. Joint work with Maciej Zworski, Izak Oltman, Martin Vogel, and Mengxuan Yang

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The Quantum Chaos Conjecture has long captivated the scientific community, proposing a crucial spectral phase transition demarcating integrable systems from chaotic systems in quantum mechanics. In integrable systems, eigenvectors typically exhibit localization with local eigenvalue statistics adhering to the Poisson distribution. In contrast, chaotic systems are characterized by delocalized eigenvectors, and their local eigenvalue statistics reflect the Sine kernel distribution reminiscent of the conventional random matrix ensembles GOE/GUE. Similarly, the Anderson conjecture reveals comparable phenomena in the context of disordered systems. This talk delves into the heart of this phenomenon, presenting a novel approach through the lens of random matrix models. By utilizing these models we aim to provide a clear and intuitive demonstration of the same phenomenon shedding light on the intricacies of these long-standing conjectures.

I will talk about the lattice Anderson model (i.e., the random Schrödinger operator of Laplacian plus i.i.d. potential), which is widely used to understand the conductivity of materials in condensed matter physics. A key phenomenon is Anderson localization, which was rigorously established in the 1980s, with one remaining question being the Bernoulli potential case. A continuous-space analog of this problem was solved by Bourgain and Kenig; and more recently, the 2D lattice setting was solved by Ding and Smart. I will discuss how we further proved the 3D lattice Anderson-Bernoulli localization near spectrum edges, based on the framework developed in these works. I will focus on the extra difficulties in the 3D lattice setting, and explain our main contribution, which is the development of a 3D discrete unique continuation principle. This is joint work with Linjun Li.

We discuss time evolution of nonlinear random and quasi-periodic systems on the lattice and elaborate on their similarities.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Questions about the properties of products of random matrices depending on a parameter naturally arise in the theory of non-self adjoint random Schrödinger operators on a strip. I'll review several old and recent results concerning such products in the case of self- and non-self adjoint random Schrödinger operators on a strip and state some open questions.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.