RAW Graduate School 2022 - Titles and Abstracts

Mini-Courses

Eman Hamza

Title : “Unitary models and quantum random walks”

Abstract : We will review the spectral theorem for unitary operators and introduce Random Quantum Walks as the main example of such operators. We will then discuss their spectral and dynamical properties.

Peter Hislop

Title : “Eigenvalue statistics for random band matrices and random Schrodinger operators”

Abstract : TBA

Mostafa Sabri :

Title : "Aspects of Delocalization"

Abstract : Delocalization is understood spectrally through the presence of absolutely continuous spectrum, dynamically through the spreading of the wavefunctions of the system with time, and spatially through the behavior of the (generalized) eigenfunctions, namely them having large support, or in ideal scenarios, being equidistributed in space. We will give an overview of these three aspects (spectral, dynamical, spatial) and illustrate them with basic examples.

This minicourse will try to be accessible without prerequisites, but it is recommended to have a basic knowledge of spectral theory, for example by watching last year's recording of the minicourse by Martin Vogel for the RAW school of 2021.

Panorama Talks :

Hakim Boumaza

Title : “Localization for one and quasi-one dimensional models”

Abstract : I will present several ergodic families of random operators in dimension one for which we prove (or expect to prove) Anderson localization. These operators are either discrete or continuous. First I discuss how to reduce the question of Anderson localization in dimension 1 to the study of an algebraic object, the Fürstenberg group. For this purpose, I present typical objects of the dimension 1: the transfer matrices, the Lyapunov exponents and a bit of Kotani theory. Then I present the tools from Lie group theory which allows to prove the required properties of the Fürstenberg group in different settings : discrete or continuous, scalar or matrix valued, Schrödinger type with Anderson potential, unitary or Dirac type operators.

Francois Huveneers

Title : "Many-body localization: A challenge for physicists and mathematician"

Abstract : Many-body localization (MBL) is an exciting field that emerged in the mid 2000s. MBL was initially understood as the generalization of Anderson localization to systems where interactions among the quantum degrees of freedom are properly taken into account. It has now been realized that MBL actually provides the best (only?) example of robust ergodicity breaking, i.e. ergodicity breaking for non-integrable systems. In striking contrast to Anderson localization, there is basically no mathematically rigorous result on MBL so far. In this talk, I will introduce the basic phenomenology. Then I will critically go through some claims from the physics literature and try to distinguish what could reasonably lead to mathematical theorems. This approach will eventually reveal some interesting features of the MBL physics, in particular of the transition, and we will see that MBL is very different from Anderson localization in many respects.

Christophe Sabot :

Title : "Reinforced random walks"

Abstract : TBA

Talks :

Davide Macera

Title : "Dynamical decay for one-dimensional Anderson eigenfunctions: quenched regime vs annealed regime"

Abstract : Motivated by the works by Jitomirskaya, Krüger, Liu, Ge, Zhao ([JK], [JKL], [GZ]), we study a disorder average of the eigenfunction correlator of the one dimensional Anderson model introduced for the Almost-Mathieu operator in [JK]. In [JKL] the authors prove that for the supercritical Almost-Mathieu Hamiltonian, this quantity coincides with its quenched counterpart (the Lyapunov exponent of the transfer cocycle) and ask whether the same is true for the Anderson model. However, the answer to this question turns out to be negative. For example, if we take the potential to be an i.i.d. sequence of random variables taking the value zero with positive probability (e.g. Bernoulli) multiplied by a coupling parameter a, then the averaged dynamical decay remains bounded in a, while the Lyapunov exponent grows like log(a). After reviewing some background notions on dynamical localisation and eigenfunction correlators, in this talk I’ll sketch a proof of the above statement. If time permits, at the end of the talk I’ll briefly discuss how to eliminate the assumption that the distribution of the potential has an atom at 0.

This is a joint work with Sasha Sodin (QMUL).

References

[GZ] Exponential dynamical localization in expectation for the one dimensional Anderson model. Journal of Spectral Theory, 10(3), 2020.

[JK] Jitomirskaya, S.; Krüger, H. Exponential Dynamical Localization for the Almost Mathieu Operator, Communications in Mathematical Physics, 322 (3), 2013.

[JKL] Jitomirskaya, S.; Krüger, H.; Liu, W. Exact dynamical decay rate for the almost Mathieu operator, Mathematical Research Letters, 27 (3), 2020.

Chokri Manai

Title : "Recent Results on Quantum Spin Glasses"

Abstract : In the past few decades, the theory of spin glasses has become a major field of interest in condensed matter physics, mathematical physics and proabbility theory. While in the classical spin glass theory many problems remain unsolved, at least a rough understanding of the underlying physics has been established. The milestone so far has been the derivation of a formula for the free energy in the classical Sherrington-Kirkpatrick model by this year's Nobel price winner Giorgio Parisi based on his replica method and its rigorous proof by Guerra and Talagrand.

The situation is vastly different for quantum spin glasses, where quantum effects are for example incorporated via a transversal field. In this case no closed formula has been found for the Quantum SK-model; and most physical results are based on numerical simulations. In this talk, I will give an introduction to the topic of quantum spin glasses. I will discuss recent results on hierarchical Quantum spin glasses, where one can rigorously prove a formula for the free energy. Moreover, we will have a quick look on the Quantum SK model, where one can at least prove the existing of replica symmetry breaking.

Daniel Sanchez-Mendoza

Title : "Principal Eigenvalue and Landscape Function of the Anderson Model on a Large Box."

Abstract : We state a precise formulation of a conjecture concerning the product of the principal eigenvalue and the sup-norm of the landscape function of the Anderson model restricted to a large box. We first provide the asymptotic of the principal eigenvalue as the size of the box grows and then use it to give a partial proof of the conjecture. For the one dimensional case we give a complete proof.