I will start by reminiding what are random walk operators on discrete group, and what are their "spectral properties". In particular we will see how are so-called Novikov-Shubin invariants defined, and what are the possible "types" of spectral measures. The aim of the talk will be to describe two joint results with B. Virag - 1) Novikov-Shubin invariants don't have to be positive, contrary to a conjecture of J. Lott and W. Lueck. 2) It can happen that for a given group two random walk operators (defined for different generating sets) have different spectral measure types (i.e. one is pure-point, the other is singular continuous.) In particular I'll try to explain why both results follow from computations done by mathematical physicists.

*Łukasz Grabowski, University of Oxford *