---
abstract: 'We propose a simplified version of the Multi-Scale Analysis of tight-binding Anderson models with strongly mixing random potentials which leads directly to uniform exponential bounds on decay of eigenfunctions in arbitrarily large finite subsets of a lattice. Naturally, these bounds imply also dynamical localization and exponential decay of eigenfunctions on the entire lattice.'
address: |
    Département de Mathématiques\
    Université de Reims, Moulin de la Housse, B.P. 1039\
    51687 Reims Cedex 2, France\
    E-mail: victor.tchoulaevski@univ-reims.fr
author:
- Victor Chulaevsky
title: |
    Direct Scaling Analysis\
    of localization in disordered systems.\
    I. Single-particle models 
---

Introduction.  {#sec:intro}
==============

In this paper, we study spectral properties of random lattice Schrödinger operators (LSO) in the framework of the Multi-Scale Analysis (MSA) developed in pioneering works [@FS83; @FMSS85; @DK89; @DK91]. Traditionally, the MSA is applied first to the resolvents in finite subsets of growing size, usually balls $\ball_{L_k}(u)$ of radius $L_k = (L_0)^{\alpha^k}$, $\alpha>1$, $k \ge 0$. The bounds on the kernels $G_{\ball_{L_k}(u)}(x,y; E)$, obtained by scale induction, are used then to derive an exponential decay of the eigenfunctions in $\DZ^d$. They can be re-used again to obtain the dynamical localization bounds.

In contrast with the MSA, a more recent method developed by Aizenman 