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

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.

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.

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.

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.

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

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

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!

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.

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.

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.

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.

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).

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 $E$.

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.

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.