  We derive finite-size scaling formulae for four-dimensional Higgs-Yukawa
models near the Gaussian fixed point. These formulae will play an essential
role in future, detailed investigation of such models. In particular, they can
be used to determine the nature of the observed phase transitions, and confirm
or rule out the possibility of having non-trivial fixed points in the
Higgs-Yukawa models. Our scaling formula for Binder's cumulant is tested
against lattice simulations carried out at weak couplings, and good agreement
is found. As a separate project, we also present preliminary results from our
study of a chirally-invariant Higgs-Yukawa model including a dimension-six
operator at finite temperature. Our work provides first indications of
first-order temperature-induced phase transitions near the infinite cutoff
limit in this model.
