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We analyze scaling functions in the $3$-$d$, $Z(2)$, $O(2)$ and $O(4)$universality classes and their finite size dependence using Monte Carlosimulations of improved $\phi^4$ models. Results for the scaling functions arefitted to the Widom-Griffiths form, using a parametrization also used inanalytic calculations. We find good agreement on the level of scaling functionsand the location of maxima in the universal part of susceptibilities. We alsofind that an earlier parametrization of the $O(4)$ scaling function, using 14parameters, is well reproduced when using the Widom-Griffiths form with onlythree parameters. We furthermore show that finite size corrections to thescaling functions are distinctively different in the $Z(2)$ and $O(N)$universality classes and determine the volume dependence of the peak locationsin order parameter and mixed susceptibilities.

Scaling functions of the three-dimensional $Z(2)$, $O(2)$ and $O(4)$ models and their finite size dependence in an external field