Consider a classical lattice model on $\mathbb{Z}^d$ and suppose that the system undergoes a phase transition as you lower the temperature, i.e., increase $\beta$. The most general definition of a first-order (discontinuous) transition (as far as I know) is that it is discontinuous if (1) the Gibbs measure (satisfying the DLR condition) is non-unique at criticality $\beta_c$. Conversely, it is continuous if the Gibbs state is unique at criticality.
I'm curious why this is equivalent to other definitions of continuous/discontinuous phase transitions, such as (2) the correlation length diverges/finite and (3) the magnetization is zero/nonzero.
Indeed, I know that for the q-states Potts model on $\mathbb{Z}^2$, it was proven (I think by Hugo Duminil-Cupin) that (2) and (3) are equivalent. I also think there's a counter example for (3) by Aernout van enter, who shown that for a family of 2D XY spin model (not the standard XY model), the transition is discontinuous, even though by Mermin-Wagner, the magnetization must be zero. However, the model seemed relatively "unphysical".
With that in mind, my question is, are there theorems that show equivalence between definitions (1) and (2) or (1) and (3) for relatively general spin systems?
