Miloš Zahradník

An alternate version of Pirogov-Sinai theory

A new approach to the Pirogov-Sinai theory of phase transitions is developed, not employing the contour models with a parameter. The completeness of the phase diagram is proven.

THE LOW-TEMPERATURE PHASE OF KAC-ISING MODELS

We analyze the low-temperature phase of ferromagnetic Kax-Ising models in dimensionsd≥2. We show that if the range of interactions is γ−1, then two disjoint translation-invariant Gibbs states exist if the inverse temperature β satisfies β−1⩾γN, where κ=d(1−ɛ)/(2d+2)(d+1), for any ε>0. The proof 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.

A Simple Inductive Approach to the Problem of Convergence of 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–Kotecký–Preiss criterion and formulate a generalization usable for situations where a successive expansion of the partition function has to be used.

Analyticity of low-temperature phase diagrams of lattice spin models

The analyticity of all strata of the Pirogov-Sinai phase diagram is proved. As a byproduct of the method, a characterization of typical volumes for which the complex partition function vanishes is given, for a Hamiltonian that is a perturbation of the real-valued one, near the point of a phase transition.

Extension of the Pirogov-Sinai theory to a class of quasiperiodic interactions

We extend the Pirogov-Sinai theory in such a manner that it applies to a large class of models with small quasiperiodic interactions as perturbations of periodic ones. We find general diophantine conditions on the frequency module of the quasiperiodic interactions and derivability conditions on the interaction potentials. These conditions allow to prove that the low temperature phase diagram is a homeomorphic deformation of the phase diagram at zero temperature.

Entropy-Driven Phase Transition in a Polydisperse Hard-Rods Lattice System

We study a system of rods onℤ2, with hard-core exclusion. Each rod has a length between 2 and N. We show that, when N is sufficiently large, and for suitable fugacity, there are several distinct Gibbs states, with orientational long-range order. This is in sharp contrast with the case N = 2 (the monomer-dimer model), for which Heilmann and Lieb proved absence of phase transition at any fugacity. This is the first example of a pure hard-core system with phases displaying orientational order, but not translational order; this is a fundamental characteristic feature of liquid crystals.

On the Ising model with strongly anisotropic external field

In this paper we analyze the equilibrium phase diagram of the two-dimensional ferromagnetic n.n. Ising model when the external field takes alternating signs on different rows. We show that some of the zero-temperature coexistence lines disappear at every positive sufficiently small temperature, whereas one (and only one) of them persists for sufficiently low temperature.

Nonperiodic long-range order for fast-decaying interactions at positive temperatures

We present the first example of an exponentially decaying interaction which gives rise to nonperiodic long-range order at positive temperatures.

ON ENTROPIC REPULSION IN LOW TEMPERATURE ISING MODELS

We investigate the phenomenon of “entropic repulsion” of interfaces in the low temperature Ising model. We study the Gibbs states in the half-space of Z d , d ≥ 3, analogously to the simplified situations of the solid-on-solid model (Bricmont, Mellouki and Frohlich 1986) and the “interface model” (Melichercik 1991). Our method is based on the Pirogov-Sinai theory and on a detailed study of cluster expansions. The crucial step consists in an estimate of some “metastable” partition sums corresponding to different heights of the interface. We formulate the basic estimate (2) and indicate its proof here.