  We develop a theory of the critical point of the ferromagnetic Ising model,
whose basic objects are the ergodic (pure) states of the infinite system. It
proves the existence of anomalous critical fluctuations, for dimension $\nu=2$
and, under a standard assumption, for $\nu=3$, for the model with nearest
neighbor interaction, in a way which is consistent with the probabilistic
approach of Cassandro, Jona-Lasinio and several others, reviewed in
Jona-Lasinio's article in Phys. Rep. 352, 439 (2001). We propose to single out
the state at the critical temperature $T_{c}$ among the ergodic thermal states
associated to temperatures $0 \le T \le T_{c}$, by a condition of non-summable
clustering of the connected two-point function. The analogous condition for the
connected $(2r)-$ point functions for $r \ge 2$, together with a scaling
hypothesis, proves that the (macroscopic) fluctuation state is quasi-free, also
at the critical temperature, after a proper rescaling, also at the critical
temperature, in agreement with a theorem by Cassandro and Jona-Lasinio, whose
proof is, however, shown to be incomplete. Other subjects include topics
related to universality, including spontaneous breaking of continuous
symmetries and violations of universality in problems of energetic and dynamic
stability.
