prof. Sadel's seminars at SISSA

[Emanuele Tuillier Illingworth]

SEMINAR ANNOUNCEMENTS


SISSA, Friday 3rd of July 11.00-12.00 room 136
Speaker: C. Sadel, Institute of Science and Technology (IST) Austria
Title:  Limiting SDEs for products of random matrices in a critical 
scaling and GOE statistics for the Anderson model on long boxes.
joint work with Balint Virag

Abstract: We consider a distribution of random matrices that are small 
perturbations of a fixed, non-random matrix T. The random part is 
coupled with a small parameter.
Then we take i.i.d. copies and study the products of these random 
matrices in a critical scaling limit where the coupling parameter goes 
to zero and the number of factors to infinity.
Using adequate Schur complements and subtracting fast rotations, one can 
describe the behaviour of the random products by Stochastic differential 
equations (SDE) in the limit.
This can be used to study limiting statistics for the eigenvalues of 
random Schrödinger operators on strips.



SISSA, Wednesday 8th of July 2.30 pm 136
Speaker: C. Sadel, Institute of Science and Technology (IST) Austria
Title: On complex analytic one-frequency coccyges. Joint work with A. 
Avila and S. Jitomirskaya

Abstract: a complex analytic, one-frequency cocycle is an analytic map 
of the trivial complex-d-dimensional vector bundle of the 
one-dimensional Torus into itself, where the base dynamics is a simple 
rotation on the torus and the action on the vector space is linear. The 
latter part can be interpreted as an analytic map A(x) from the torus 
into the d x d matrices and the rotation part gives the frequency alpha.
Associated to the iterates of the cocycle are d Lyapunov exponents. We 
prove joint-continuity in (alpha,A) of all the Lyapunov exponents at 
irrational frequencies alpha, give a criterion for domination and prove 
that a dense open subset of cocycles are dominated. These results 
generalize known results and methods for cocycles of SL(2) matrices.
The subject got attention  in Mathematical Physics as such cocycles 
appear for the transfer matrix process of certain quasi-periodic, 
one-dimensional Schrödinger operators and the energies in the spectrum 
are characterized by having a non-dominated cocycle. This could be a 
first step towards a Ten Martini theorem (Cantor spectrum) for generic 
quasi-periodic Schrödinger operators on strips.
Also, A. Avila's global theory for quasiperiodic operators on L^2(Z) is 
based on these methods and might be generalized with these methods.