  We construct exact solutions to the functional renormalisation group equation
of the O(N) model and the Gross-Neveu model at large N for $2<d<4$, without
specifying the form of the regulator. This allows to investigate which
quantities are independent of the choice of regulator without being plagued by
truncation artefacts. We find that only universal quantities, like critical
exponents, and qualitative features, like the existence of a finite vacuum
expectation value, are regulator-independent, whereas values of coupling
constants are generically arbitrary. We also provide a general algorithm to
construct a concrete operator basis for truncations in the derivative expansion
and the Blaizot-M\'endez-Wschebor scheme.
