A simple inductive approach to the problem of convergenceof cluster expansions of polymer models

We explain a simple inductive method for the analysis of the convergence of cluster expansions (Taylor expansions, Mayer expansions) for the partition functions of "polymer models". We give a very simple proof of the "Dobrushin-Koteck'y-Preiss criterion" and formulate its generalization usable for situations where a successive expansion of the partition function has to be used

Non-periodic long-range order for fast decaying interactions at positive temperatures

We present the first example of an exponentially decaying interaction which gives rise to non-periodic long-range order at positive temperatures.Comment: 7 pages, Late

On the Gibbs phase rule in the Pirogov-Sinai regime

We consider extended Pirogov-Sinai models including lattice and continuum particle systems with Kac potentials. Calling λ an intensive variable conjugate to an extensive quantity α appearing in the Hamiltonian via the additive term -λα, we prove that if a Pirogov-Sinai phase transition with order parameter λ occurs at λ = 0, then this is the only point in an interval of values of λ centered at 0, where phase transitions occur

The low-temperature phase of Kac-Ising models

We analyse the low temperature phase of ferromagnetic Kac-Ising models in dimensions $d\geq 2$. We show that if the range of interactions is \g^{-1}, then two disjoint translation invariant Gibbs states exist, if the inverse temperature \b satisfies \b -1\geq \g^\k where \k=\frac {d(1-\e)}{(2d+1)(d+1)}, for any \e>0. The prove involves the blocking procedure usual for Kac models and also a contour representation for the resulting long-range (almost) continuous spin system which is suitable for the use of a variant of the Peierls argument.Comment: 19pp, Plain Te