  We introduce an approach to find approximate numerical solutions of truncated
bootstrap equations for Conformal Field Theories (CFTs) in arbitrary
dimensions. The method is based on a stochastic search via a Metropolis
algorithm guided by an action $S$ which is the logarithm of the truncated
bootstrap equations for a single scalar field correlator. While numerical
conformal bootstrap methods based on semi-definite programming put rigorous
exclusion bounds on CFTs, this method looks for approximate solutions, which
correspond to local minima of $S$, when present, and can be even far from the
extremality region. By this protocol we find that if no constraint on the
operator scaling dimensions is imposed, $S$ has a single minimum, corresponding
to the Free Theory. If we fix the external operator dimension, however, we
encounter minima that can be studied with our approach. Imposing a conserved
stress-tensor, a $\mathbf{Z}_2$ symmetry and one relevant scalar, we identify
two regions where local minima of $S$ are present. When projected in the
$(\Delta_\sigma, \Delta_{\epsilon})$-plane, $\sigma$ and $\epsilon$ being the
external and the lightest exchanged operators, one of these regions essentially
coincides with the extremality line found in previous bootstrap studies. The
other region is along the generalized free theories in $d = 2$ and below that
in both $d = 3$ and $d = 4$. We empirically prove that some of the minima found
are associated to known theories, including the $2d$ and $3d$ Ising theories
and the $2d$ Yang-Lee model.
