Scheme independence at first order phase transitions and the renormalisation group

We analyse approximate solutions to an exact renormalisation group equation with particular emphasis on their dependence on the regularisation scheme, which is kept arbitrary. Physical quantities related to the coarse-grained potential of scalar QED display universal behaviour for strongly first-order phase transitions. Only subleading corrections depend on the regularisation scheme and are suppressed by a sufficiently large UV scale. We calculate the relevant coarse-graining scale and give a condition for the applicability of Langer's theory of bubble nucleation.