Multi-Particle Anderson Localisation: Induction on the Number of Particles

This paper is a follow-up of our recent papers [CS08] and [CS09] covering the two-particle Anderson model. Here we establish the phenomenon of Anderson localisation for a quantum N -particle system on a lattice Z with short-range interaction and in presence of an IID external potential with sufficiently regular marginal cumulative distribution function (CDF). Our main method is an adaptation of the multi-scale analysis (MSA; cf. [FS], [FMSS], [DK]) to multi-particle systems, in combination with an induction on the number of particles, as was proposed in our earlier manuscript [CS07]. Similar results have been recently obtained in an independent work by Aizenman and Warzel [AW08]: they proposed an extension of the Fractional-Moment Method (FMM) developed earlier for single-particle models in [AM93] and [ASFH01] (see also references therein) which is also combined with an induction on the number of particles. An important role in our proof is played by a variant of Stollmann’s eigenvalue concentration bound (cf. [St00]). This result, as was proved earlier in [C08], admits a straightforward extension covering the case of multi-particle systems with correlated external random potentials: a subject of our future work. We also stress that the scheme of our proof is not specific to lattice systems, since our main method, the MSA, admits a continuous version. A proof of multi-particle Anderson localization in continuous interacting systems with various types of external random potentials will be published in a separate papers.