Is there any proof of the existence of phase transition in models of statistical mechanics of the Ising type models without using the Peierls' argument and its variations?
By models of the Ising type I want to mean Ising models of first and seconds neighbors over lattices $Z^d$ with $d\geq 2$.
wsc's answer (i.e., Onsager's computation of the free energy) provides one alternative road to a proof of a phase transition in the Ising model. It implies the existence of a phase transition in dimension 2 (for the nearest-neighbor model). Combined with correlation inequalities, this implies existence of a phase transition in any dimension d≥2, and interactions of any range (provided that nearest-neighbors also interact).
An alternative approach is via reflection positivity, see, for example, these lecture notes. In particular Theorem 3.5 therein implies the existence of a phase transition in all $O(n)$ models in dimensions $3$ and more. This approach (via the so-called infrared bound) cannot yield the result in dimension $2$, however (but notice that the $O(n)$ model do not display spontaneous magnetization in dimension 2 when $n\geq 2$).
You can also use slight variants of the Peierls approach, using percolation arguments. For example, you can use a comparison (relying on the FKG inequality available in the random-cluster representation of the Ising model) with the percolation model, in order to show that the existence of a phase transition in the percolation model implies the existence of a phase transition in the Ising model. However, to prove the former, you use again a Peierls-type argument...
The $d=2$ (square lattice) Ising model has a special "duality" property (the high-temperature and low-temperature partition functions can be mapped on to one another) discovered by Kramers & Wannier in 1941. This doesn't rigorously prove that a phase transition exists, but it remarkably predicts the critical point where a phase transition, if it did exist, would have to occur. Shortly thereafter, Onsager solved the model exactly (1944) thereby proving the phase transition exists.
For $d \ge 4$ mean field theory of the hypercubic-lattice Ising model is exact. Mean field theory, of course, also predicts a phase transition.
$d=3$ doesn't seem to admit an exact solution, but can be solved numerically, and displays a phase transition that can be characterized by critical exponents that are not mean-field.
All that said, what do you have against the Peierls' argument? It's a really powerful and beautiful thing...
