Motivated by a classical result of Sz.-Nagy in functional analysis, Dixmier asked in 1950 which group representations can be made unitary. This question is still open, but I will report on some recent progress. We approach the question with ideas borrowed from XIXth century electricity theory as well as from contemporary percolation theory. As a result, we obtain notably non-unitarizable representations for Burnside groups and a new characterization of amenable groups. (The talk will be expository.)