  Massive and massless Gaussian free fields can be described as generalized
Gaussian processes indexed by an appropriate space of functions. In this
article we study various approaches to approximate these fields and look at the
fractal properties of the thick points of their cut-offs. Under some sufficient
conditions for a centered Gaussian process with logarithmic variance we study
the set of thick points and derive their Hausdorff dimension. We prove that
various cut-offs for Gaussian free fields satisfy these assumptions. We also
give sufficient conditions for comparing thick points of different cut-offs.
