Efficient Anderson localization bounds for large multi-particle systems

We study multi-particle interactive quantum disordered systems on a polynomially growing countable connected graph (Z, Ε). The main novelty is to give localization bounds uniform in finite volumes (subgraphs) in Z^N as well as for the whole of Z^N. Such bounds are proved here by means of a comprehensive fixed-energy multi-particle multi-scale analysis. We consider - for the first time in the literature - a discrete N-particle model with an infinite-range, sub-exponentially decaying interaction, and establish (1) exponential spectral localization, and (2) strong dynamical localization with sub-exponential rate of decay of the eigenfunction correlators with respect to the natural symmetrized distance in the multi-particle configuration space.