" Motivated by a classical result of Szökefalvi-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 progress obtained with Epstein and Ozawa. 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 amenability. The talk will be expository."