Some Uniform Estimates in Products of Random Matrices

We provide a complete and self-contained proof of spectral and dynamical localization for the one-dimensional Anderson model, starting from the positivity of the Lyapunov exponent provided by Fürstenberg's theorem. That is, a Schrödinger operator in ℓ 2 (Z) whose potential is given by independent identically distributed (i.i.d.) random variables almost surely has pure point spectrum with exponentially decaying eigenfunctions and its unitary group exhibits exponential off-diagonal decay, uniformly in time. This is achieved by way of a new result: for the Anderson model, one typically has Lyapunov behavior for all generalized eigenfunctions. We also explain how to obtain analogous statements for extended CMV matrices whose Verblunsky coefficients are i.i.d., as well as for half-line analogs of these models.V.B., D.D., V.G., T.

show abstract

“…random matrices [37] and one-parameter families of i.i.d. matrices [50]. However, we wish to emphasize that this proof of the LDT is new.…”

Section: 4mentioning

confidence: 95%

Localization for the one-dimensional Anderson model via positivity and large deviations for the Lyapunov exponent

Bucaj

Damanik

Fillman

et al. 2019

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…These LDT type estimates for }T ra,bs,ω,z } were proved in [25]. Here we use a matrix-entry version from [26,Theorem 5]. The result was proved for SLp2, Rqcocycle.…”

Section: 4mentioning

confidence: 96%

“…(1) Equation ( 7.4) should be φ β,γ ω,ra,bs pzq " We believe this can be used then to derive (4.9) by [26,Theorem 5]. Then one could complete the proof of double elimination in [11].…”

Section: B2mentioning

confidence: 99%

Localization for random CMV matrices

Zhu¹

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Here we will use a corresponding statement for the matrix elements [16] Lemma 2.3 ( "uniform-LDT"). For any ǫ ą 0, there exists η " ηpǫq ą 0 such that, there exists N 0 " N 0 pǫq, such that for every b ´a ą N 0 , and any E in a compact set,…”

Section: ṫHementioning

confidence: 99%

Large Deviations of the Lyapunov Exponent and Localization for the 1D Anderson Model

Jitomirskaya

Zhu

2019

Commun. Math. Phys.

View full text Add to dashboard Cite

The proof of Anderson localization for the 1D Anderson model with arbitrary (e.g. Bernoulli) disorder, originally given by Carmona-Klein-Martinelli in 1987, is based in part on the multi-scale analysis. Later, 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. In this paper we present a short proof along these lines, for the Anderson model. To prove dynamical localization we also develop a uniform version of Craig-Simon's bound that works in high generality and may be of independent interest.

show abstract

Some Uniform Estimates in Products of Random Matrices

Cited by 12 publications

References 7 publications

Localization for the one-dimensional Anderson model via positivity and large deviations for the Lyapunov exponent

Localization for the one-dimensional Anderson model via positivity and large deviations for the Lyapunov exponent

Localization for random CMV matrices

Large Deviations of the Lyapunov Exponent and Localization for the 1D Anderson Model

Contact Info

Product

Resources

About