# Schedule for: 19w5086 - Random Matrix Products and Anderson Localization

Beginning on Sunday, September 15 and ending Friday September 20, 2019

All times in Banff, Alberta time, MDT (UTC-6).

Sunday, September 15 | |
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16:00 - 17:30 | Check-in begins at 16:00 on Sunday and is open 24 hours (Front Desk - Professional Development Centre) |

17:30 - 19:30 |
Dinner ↓ A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |

20:00 - 22:00 | Informal gathering (Corbett Hall Lounge (CH 2110)) |

Monday, September 16 | |
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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) |

09:00 - 10:00 |
Tom VandenBoom: The 1D Anderson Model via positive Lyapunov Exponents and a LDT ↓ We discuss a simple proof of localization for the 1D discrete Anderson model. While the result itself is well-known, our strategy is broadly applicable in the 1D setting: use positive Lyapunov exponents and a Large Deviation Theorem to demonstrate generalized eigenfunction decay at an initial scale, then apply the Avalanche Principle in the absence of double resonances to get Anderson localization. In this talk we advertise the general strategy and discuss the elimination of double resonances in the i.i.d. random setting. (TCPL 202) |

10:00 - 10:30 | Coffee Break (TCPL Foyer) |

10:30 - 11:30 |
Victor Kleptsyn: Furstenberg theorem: now with a parameter! ↓ The classical Furstenberg theorem describes the (almost sure) behaviour of a random product of independent matrices; their norms turn out to grow exponentially. In our joint work with A. Gorodetski, we study what happens if the random matrices depend on an additional parameter. It turns out that in this new situation, the conclusion changes. Namely, under some conditions, there almost surely exists a (random) "exceptional" set on parameters where the lower limit for the Lyapunov exponent vanishes. Our results are related to the Anderson localization in dimension one, providing a purely dynamical viewpoint on its proof. I will also speak about some generalizations and related open questions. (TCPL 202) |

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) |

13:00 - 14:00 |
Guided Tour of The Banff Centre ↓ Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus. (Corbett Hall Lounge (CH 2110)) |

14:00 - 14:20 |
Group Photo ↓ Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo! (TCPL 201) |

14:20 - 15:00 |
Xiaowen Zhu: A short proof of Anderson localization for the 1-d Anderson model ↓ The proof of Anderson localization for 1D Anderson model with arbitrary (e.g. Bernoulli) disorder, originally given by Carmona-Klein-Martinelli in 1987, is based on the Furstenberg theorem and multi-scale analysis. This topic has received a renewed attention lately, with two recent new proofs, exploiting the one-dimensional nature of the model. At the same time, in the 90s it was realized that for one-dimensional models with positive Lyapunov exponents some parts of multi-scale analysis can be replaced by considerations involving subharmonicity and large deviation estimates for the corresponding cocycle, leading to nonperturbative proofs for 1D quasiperiodic models. Here we present a proof along these lines, for the Anderson model. We also include a proof of dynamical localization based on the uniform version of Craig-Simon that works in high generality and may be of independent interest. It is a joint work with S. Jitomirskaya. Our entire proof of spectral localization fits in three pages and we expect to present almost complete detail during the talk. (TCPL 201) |

15:00 - 15:30 | Coffee Break (TCPL Foyer) |

15:30 - 16:30 |
Peter Baxendale: Random matrix products and random dynamical systems ↓ The talk will be in two parts. The first part will be tutorial in nature, discussing the connection between random matrix products and random dynamical systems. The second part will show how some of these ideas are used in the analysis of a stochastic bifurcation scenario for a damped and random excited non-linear harmonic oscillator. (TCPL 201) |

16:30 - 17:30 | Problem Session I (TCPL 201) |

17:30 - 19:30 |
Dinner ↓ A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |

Tuesday, September 17 | |
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07:00 - 09:00 | Breakfast (Vistas Dining Room) |

09:00 - 10:00 |
Ilya Goldsheid: Products of non-stationary Markov dependent matrices: positivity of the top Lyapunov exponent ↓ Consider a sequence of matrices from $SL(m,R)$ of the form $g_n=F(x_n)$, where $x_n$ is a non-homogeneous Markov chain with phase space $X$ and $F: X\mapsto SL(m,R)$. I shall discuss conditions under which the norm of the product $g_ng_{n-1}...g_1$ of such matrices grows exponentially as $n\to \infty$. The main result generalizes the well known theorem proved by A. Virtser in 1980 who considered the case of stationary Markov chains. (TCPL 202) |

10:00 - 10:30 | Coffee Break (TCPL Foyer) |

10:30 - 11:30 |
Yuki Takahashi: Diophantine property of matrices and absolute continuity of the Furstenberg measure ↓ We prove that almost every finite collection of matrices in $GL_d(\mathbb{R})$ and $SL_d(\mathbb{R})$ with positive entries is Diophantine. This immediately implies that the associated Furstenberg measure has the "expected dimension" (joint work with B. Solomyak). We then show that the Furstenberg measure can be absolutely continuous even when the generating matrices are not uniformly hyperbolic (joint work with R. Tanaka, in progress). (TCPL 202) |

11:30 - 13:30 | Lunch (Vistas Dining Room) |

13:30 - 14:15 |
Florian Dorsch: Random perturbations of hyperbolic dynamics ↓ A sequence of large invertible matrices given by a small random perturbation around a fixed diagonal and positive matrix induces a random dynamics on a high-dimensional sphere. For a certain class of rotationally invariant random perturbations it is shown that the dynamics approaches the stable fixed points of the unperturbed matrix up to errors even if the strength of the perturbation is large compared to the relative increase of nearby diagonal entries of the unperturbed matrix specifying the local hyperbolicity. The talk is based on a joint work with Hermann Schulz-Baldes. (TCPL 201) |

14:15 - 15:00 |
Nishant Rangamani: Exponential Dynamical Localization for Random Word Models ↓ We give a new proof of spectral localization for one-dimensional Schrodinger operators whose potentials arise by randomly concatenating words from an underlying set. We then show that once one has the existence of a complete orthonormal basis of eigenfunctions (with probability one), the same estimates used to prove it naturally lead to a proof of exponential dynamical localization in expectation (EDL) on any compact interval not containing a finite set of critical energies. (TCPL 201) |

15:00 - 15:30 | Coffee Break (TCPL Foyer) |

15:30 - 16:30 |
Jake Fillman: Sufficient criteria for the application of Fürstenberg's theorem with applications to the 1D continuum Bernoulli Anderson model ↓ Almost every proof of localization for one-dimensional Anderson models begins with the classical Fürstenberg theorem about products of random matrices. We review the statement of Fürstenberg's theorem and some sufficient conditions which imply the hypotheses of said theorem. As an application, we give a simple proof of positive Lyapunov exponents in the 1D continuum Bernoulli Anderson model. (TCPL 201) |

16:30 - 17:30 | Problem Session II (TCPL 201) |

17:30 - 19:30 | Dinner (Vistas Dining Room) |

Wednesday, September 18 | |
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07:00 - 09:00 | Breakfast (Vistas Dining Room) |

09:00 - 10:00 |
Charles Smart: Unique continuation and localization on the planar lattice ↓ I will discuss joint work with Jian Ding in which we establish localization near the edge for the Anderson Bernoulli model on the two dimensional lattice. Our proof follows the program of Bourgain--Kenig and uses a new unique continuation result inspired by Buhovsky--Logunov--Malinnikova--Sodin. (TCPL 202) |

10:00 - 10:30 | Coffee Break (TCPL Foyer) |

10:30 - 11:30 |
Victor Kleptsyn: Non-stationary versions of Anderson Localization and Furstenberg Theorem on random matrix products ↓ We consider 1D discrete Schr\"odinger operators with bounded random but not necessarily identically distributed values of the potential. The distribution at a given site is not assumed to be absolutely continuous (or to contain an absolutely continuous component). We prove spectral localization (with exponentially decaying eigenfunctions) as well as dynamical localization for this model.
An important ingredient of the proof is a non-stationary analog of the Furstenberg Theorem on random matrix products. (TCPL 202) |

11:30 - 13:30 | Lunch (Vistas Dining Room) |

13:30 - 17:30 | Free Afternoon (Banff National Park) |

17:30 - 19:30 | Dinner (Vistas Dining Room) |

Thursday, September 19 | |
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07:00 - 09:00 | Breakfast (Vistas Dining Room) |

09:00 - 10:00 |
Selim Sukhtaiev: Anderson localization for radial trees ↓ In this talk I will discuss Anderson localization for several continuum and discrete models on radial trees, including random branching, random length, and random Kirchhoff models. This is joint work with D. Damanik and J. Fillman. (TCPL 202) |

10:00 - 10:30 | Coffee Break (TCPL Foyer) |

10:30 - 11:00 |
Jacob Kesten: Random Hamiltonians with Arbitrary Point Interactions: Positivity of the Lyapunov Exponent I ↓ In this talk, we will consider disordered Hamiltonians given by the Laplace operator subject to arbitrary random self-adjoint singular perturbations supported on random discrete subsets of the real line. We prove positivity of the Lyapunov exponent for suitably random vertex conditions and all energies outside of a discrete set, which is used to prove Anderson Localization for such operators. This is joint work with Damanik, Fillman, and Sukhtaiev. (TCPL 202) |

11:00 - 11:30 |
Mark Helman: Random Hamiltonians with Arbitrary Point Interactions: Positivity of the Lyapunov Exponent II ↓ In this talk, we will consider disordered Hamiltonians given by the Laplace operator subject to arbitrary random self-adjoint singular perturbations supported on random discrete subsets of the real line. We prove positivity of the Lyapunov exponent for suitably random vertex conditions and all energies outside of a discrete set, which is used to prove Anderson Localization for such operators. This is joint work with Damanik, Fillman, and Sukhtaiev. (TCPL 202) |

11:30 - 13:30 | Lunch (Vistas Dining Room) |

13:30 - 14:15 |
Fernando Quintino: Phase transition of capacity for uniform \(G_{\delta}\) sets ↓ We study the capacity of a uniform \(G_{\delta}\) set. Changing the speed at which the lengths of intervals generating it decrease, we observe a sharp phase transition from full to zero capacity. Such a $G_{\delta}$ set is also interesting because it can be considered as a model case for the set of exceptional energies in the parametric version of the Furstenberg theorem. In the talk, we will demonstrate the techniques used and as well as some interesting capacity-related examples. (TCPL 201) |

14:15 - 15:00 |
Hyunkyu Jun: Cantor Spectrum for CMV and Jacobi Matrices with Coefficients arising from Generalized Skew-Shifts ↓ In a paper by Avila, Bochi, Damanik (2009), the authors consider continuous SL(2,R)-cocycles which arise from generalized skew-shifts and they prove the C^0-density of uniformly hyperbolic SL(2,R)-cocycles. Using this and “a projection lemma” they show the C^0-density of uniformly hyperbolic Schrodinger cocycles.
Our work builds upon their results on SL(2,R)-cocycles. We consider continuous cocycles arising from CMV and Jacobi matrices. Assuming the Verblunsky and Jacobi coefficients arise from generalized skew-shifts, we prove that uniform hyperbolicity of the associated cocycles is C^0-dense. This implies that the associated CMV and Jacobi matrices have Cantor spectrum for a generic continuous sampling map. (TCPL 201) |

15:00 - 15:30 | Coffee Break (TCPL Foyer) |

15:30 - 16:30 |
Ilya Goldsheid: Product of random matrices depending on a parameter revisited ↓ In this talk, I shall consider products of i.i.d. matrices $g_j(t),\ j\ge 1,$ where $t$ is a parameter, $\ t\in T,$ and $T$ is a compact metric space. Matrices $g_(\cdot)$ are continuous functions of $t$. I shall discuss necessary and sufficient conditions under which with probability 1
\[
\frac1n \ln\| g_n(t)\ldots g_1(t)\| \to \lambda(t)\ \ \text{ uniformly in $t\in T$,}
\]
where $\lambda(t)$ is the corresponding Lyapunov exponent.
I shall then explain what happens when $g_j$ are matrices corresponding to the Anderson model and the parameter is the energy $E$. (TCPL 201) |

16:30 - 17:30 | Problem Session III (TCPL 201) |

17:30 - 19:30 | Dinner (Vistas Dining Room) |

Friday, September 20 | |
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07:00 - 09:00 | Breakfast (Vistas Dining Room) |

09:00 - 10:00 |
Anton Gorodetski: Sums of Cantor sets and non-stationary Anderson-Bernoulli Model ↓ Structure of sums of Cantor sets is a classical subject in fractal geometry. We will describe some of the results and techniques used in the area, and explain how they can be used to construct 1D discrete Schrodinger operator with a random non-stationary potential such that the essential spectrum intersects an interval at a Cantor set of zero measure. The talk is based on a joint work with V.Kleptsyn. (TCPL 202) |

10:00 - 10:30 | Coffee Break (TCPL Foyer) |

11:30 - 12:00 |
Checkout by Noon ↓ 5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon. (Front Desk - Professional Development Centre) |

12:00 - 13:30 | Lunch from 11:30 to 13:30 (Vistas Dining Room) |