The Projection of some Random Cantor sets and the Decay Rate of the Favard length.

Speaker: 

Shiwen Zhang

Institution: 

Michigan State

Time: 

Friday, January 12, 2018 - 2:00pm

Location: 

Rh 340N

The Favard length of a set E has a probabilistic interpretation: up to a constant factor, it is the probability that the Buffon's needle, a long line segment dropped at random, hits E. In this talk, we study the Favard length of some random Cantor sets of dimension 1. Replace the unit disc by 4 disjoint sub-discs of radius 1/4 inside. By repeating this operation in a self-similar manner and adding a random rotation in each step, we can generate a random Cantor set D. Let D_n be the n-th generation in the construction, which is comparable to the 4^{-n}-neighborhood of D. We are interested in the decay rate of the Favard length of these sets D_n as n tends to infinity, which is the likelihood (up to a constant) that the Buffon's needle will fall into the 4^{-n}-neighborhood of D. It is well known that the lower bound for such 1-dimensional set is constant multiple of 1/n. We show that the upper bound of the Favard length of D_n is also constant multiple of 1/n in the average sense.

 

Spectral gaps for quasi-periodic Schrodinger operators with Liouville frequencies

Speaker: 

Yunfeng Shi

Institution: 

Fudan University

Time: 

Thursday, November 2, 2017 - 2:00pm

Location: 

RH 340N

Abstract: In this talk, we consider the spectral gaps of quasi-periodic Schrodinger operators with Liouville frequencies. By establishing quantitative reducibility of the associated Schrodinger cocycle,  we show that the size of the spectral gaps decays exponentially. This is a joint work with Wencai Liu. 

A new proof of Anderson localization for the 1-d Anderson model.

Speaker: 

Xhiaowen Zhu

Institution: 

UCI

Time: 

Thursday, October 19, 2017 - 2:00pm

Location: 

RH 340P

We give a new and short proof of spectral localization for the 1-d Anderson model with any disorder. The original proof was given by Carmona-Klein-Martinelli in 1987, based on multi-scale analysis. Our proof is based on the large deviation estimates and positivity and subharmonicity of the Lyapunov exponent. We also show how to improve the estimates to get a uniform and quantitative version that allows us to get the exponential dynamical localization.  It is joint work with S. Jitomirskaya. Complete details of the spectral localization proof will be presented during the talk. 

 

Pages

Subscribe to RSS - Mathematical Physics