scholarly journals Band Edge Localization Beyond Regular Floquet Eigenvalues

2020 ◽  
Vol 21 (7) ◽  
pp. 2151-2166
Author(s):  
Albrecht Seelmann ◽  
Matthias Täufer
Keyword(s):  
Floquet Theory ◽  
Large Deviation ◽  
Random Variables ◽  
Unique Continuation ◽  
Band Edge ◽  
Random Schrödinger Operators ◽  
Initial Scale ◽  
Quantitative Unique Continuation ◽  
Scale Estimate ◽  
Floquet Eigenvalues

Abstract We prove that localization near band edges of multi-dimensional ergodic random Schrödinger operators with periodic background potential in $$L^2({\mathbb {R}}^d)$$ L 2 ( R d ) is universal. By this, we mean that localization in its strongest dynamical form holds without extra assumptions on the random variables and independently of regularity or degeneracy of the Floquet eigenvalues of the background operator. The main novelty is an initial scale estimate the proof of which avoids Floquet theory altogether and uses instead an interplay between quantitative unique continuation and large deviation estimates. Furthermore, our reasoning is sufficiently flexible to prove this initial scale estimate in a non-ergodic setting, which promises to be an ingredient for understanding band edge localization also in these situations.

scholarly journals Chover-Type Laws of the Iterated Logarithm for Kesten-Spitzer Random Walks in Random Sceneries Belonging to the Domain of Stable Attraction

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Wensheng Wang ◽  
Anwei Zhu
Keyword(s):  
Random Walk ◽  
Large Deviation ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Stable Law ◽  
Iterated Logarithm ◽  
Cluster Set ◽  
Random Scenery ◽  
Unique Integer

Let X={Xi,i≥1} be a sequence of real valued random variables, S0=0 and Sk=∑i=1kXi  (k≥1). Let σ={σ(x),x∈Z} be a sequence of real valued random variables which are independent of X’s. Denote by Kn=∑k=0nσ(⌊Sk⌋)  (n≥0) Kesten-Spitzer random walk in random scenery, where ⌊a⌋ means the unique integer satisfying ⌊a⌋≤a<⌊a⌋+1. It is assumed that σ’s belong to the domain of attraction of a stable law with index 0<β<2. In this paper, by employing conditional argument, we investigate large deviation inequalities, some sufficient conditions for Chover-type laws of the iterated logarithm and the cluster set for random walk in random scenery Kn. The obtained results supplement to some corresponding results in the literature.

scholarly journals CLT-related large deviation bounds based on Stein's method

2007 ◽  
Vol 39 (3) ◽  
pp. 731-752 ◽  
Author(s):  
Martin Raič
Keyword(s):  
Random Graphs ◽  
Finite Population ◽  
Large Deviation ◽  
Random Variables ◽  
Stein’S Method ◽  
Stein's Method ◽  
Population Statistics ◽  
Sums Of Random Variables ◽  
Dependence Structures

Large deviation estimates are derived for sums of random variables with certain dependence structures, including finite population statistics and random graphs. The argument is based on Stein's method, but with a novel modification of Stein's equation inspired by the Cramér transform.

scholarly journals Large Deviation Analysis of a Droplet Model Having a Poisson Equilibrium Distribution

2015 ◽  
Vol 2015 ◽  
pp. 1-15
Author(s):  
Richard S. Ellis ◽  
Shlomo Ta’asan
Keyword(s):  
Equilibrium Distribution ◽  
Large Deviation ◽  
Random Variables ◽  
Rate Function ◽  
Number Density ◽  
Droplet Model ◽  
Empirical Measures ◽  
Deviation Analysis ◽  
Droplet Sizes ◽  
Positive Integers

In this paper we use large deviation theory to determine the equilibrium distribution of a basic droplet model that underlies a number of important models in material science and statistical mechanics. Given b∈N and c>b, K distinguishable particles are placed, each with equal probability 1/N, onto the N sites of a lattice, where K/N equals c. We focus on configurations for which each site is occupied by a minimum of b particles. The main result is the large deviation principle (LDP), in the limit K→∞ and N→∞ with K/N=c, for a sequence of random, number-density measures, which are the empirical measures of dependent random variables that count the droplet sizes. The rate function in the LDP is the relative entropy R(θ∣ρ∗), where θ is a possible asymptotic configuration of the number-density measures and ρ∗ is a Poisson distribution with mean c, restricted to the set of positive integers n satisfying n≥b. This LDP implies that ρ∗ is the equilibrium distribution of the number-density measures, which in turn implies that ρ∗ is the equilibrium distribution of the random variables that count the droplet sizes.

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