  Using the local potential approximation of the exact renormalization group
(RG) equation, we show the various domains of values of the parameters of the
O(1)-symmetric scalar Hamiltonian. In three dimensions, in addition to the
usual critical surface $S_{c}$ (attraction domain of the Wilson-Fisher fixed
point), we explicitly show the existence of a first-order phase transition
domain $S_{f}$ separated from $S_{c}$ by the tricritical surface $S_{t}$
(attraction domain of the Gaussian fixed point). $S_{f}$ and $S_{c}$ are two
distinct domains of repulsion for the Gaussian fixed point, but $S_{f}$ is not
the basin of attraction of a fixed point. $S_{f}$ is characterized by an
endless renormalized trajectory lying entirely in the domain of negative values
of the $\phi ^{4}$-coupling. This renormalized trajectory exists also in four
dimensions making the Gaussian fixed point ultra-violet stable (and the
$\phi_{4}^{4}$ renormalized field theory asymptotically free but with a wrong
sign of the perfect action). We also show that very retarded classical-to-Ising
crossover may exist in three dimensions (in fact below four dimensions). This
could be an explanation of the unexpected classical critical behavior observed
in some ionic systems.
