Functional Integration and Disordered Systems

Abstract

Very often disordered systems are self-averaging in the thermodynamic limit. This means that any physical quantity is almost surely equal to its expectation value. But the “disorder” can be seen as a random potential, i.e. a random scalar field, so that computing mean values is a field theory problem. Thus, functional integration techniques can be used in addition to probabilistic ones which usually fail when the disorder is small. Question: Can rigorous functional integration allow to get results in weakly disordered systems? For instance, is it possible to prove that electrons in a metal with few impurities have extended states, completing the proof of a metal-insulator transition in semi-conductors?