| Monday, January 15 |
| 07:00 - 08:45 |
Breakfast ↓ Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |
| 08:45 - 09:00 |
Introduction and Welcome by BIRS Staff ↓ A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions. (TCPL 201) |
| 09:10 - 09:50 |
Anton Gorodetski: On the spectrum of 1D Schrodinger operators with random noise ↓ 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. (TCPL 201) |
| 10:00 - 10:30 |
Coffee Break (TCPL Foyer) |
| 10:30 - 11:10 |
Zhenghe Zhang: Anderson localization for potentials generated by hyperbolic transformations ↓ 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. (TCPL 201) |
| 11:30 - 13:00 |
Lunch ↓ Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |
| 14:20 - 15:00 |
Peter Mueller: On the return probability of the simple random walk on Galton-Watson trees ↓ 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. (TCPL 201) |
| 15:00 - 15:30 |
Coffee Break (TCPL Foyer) |
| 15:30 - 16:10 |
Jacob Shapiro: Classification of 1D Chiral Insulators in the mobility gap regime ↓ 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. (TCPL 201) |
| 16:20 - 17:00 |
Frédéric Klopp: The ground state of a system of interacting fermions in a random field: localization, entanglement entropy ↓ 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. (TCPL 201) |
| 17:30 - 19:30 |
Dinner ↓ A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building. (Vistas Dining Room) |
