Title: Asymptotic Many-Body Localization in Ordered and Disordered Hamiltonian
lattices

Abstract:
It is known that energy transfer in quantum of classical Hamiltonian
systems can sometimes be much slowed down or even suppressed. Anderson
localization, breathers, KAM tori or Nekhoroshev estimates can in some
cases be invoked to justify this claim. However, given a chain of
oscillators at positive temperature in the infinite volume limit, it is
generally hard to infer any clear picture on heat transfer out of such
mathematical results. In this talk I will consider a class of nearly
integrable Hamiltonian chains in a weak coupling regime, sometimes
translated into a high or small temperature regime. I will present some
rigorous asymptotic results, suggesting a very rapid fall-off of the
thermal conductivity with the coupling strength (or temperature). I will
argue that both disorder (random masses for example) and strong
anharmonicity play a similar role in the considered regime. If times
allows, I will discuss the delicate difference between classical and
quantum systems, and present some conjecture on existence/non-existence of
true many-body localization.