  The autonomous renormalization of the O(N)-symmetric scalar theory is based
on an infinite re-scaling of constant fields, whereas finite-momentum modes
remain finite. The natural framework for a detailed analysis of this method is
a system of finite size, where all non-constant modes can be integrated out
perturbatively and the constant mode is treated by a saddle-point approximation
in the thermodynamic limit. The calculation provides a better understanding of
the properties of of the effective action and corroborates earlier findings
concerning a heavy Higgs particle at about 2 TeV.
