Jacek Miekisz

Breaking of Periodicity at Positive Temperatures

We discuss a classical lattice gas model without periodic or quasiperiodic ground states. The only ground state configurations of our model are nonperiodic Thue-Morse sequences. We show that low temperature phases of such models can be ordered. In fact, we prove the existence of an ordered (nonmixing) low temperature translation invariant equilibrium state which has nonperiodic Gibbs states in its extremal decomposition.

Fractal symmetry in an Ising model

Gives an example of a short-range Ising model with a unique ground state with two unusual properties: it has some continuous spectrum, and it has fractal symmetry.

A microscopic model with quasicrystalline properties

A classical lattice gas model with two-body nearest neighbor interactions and without periodic ground-state configurations is presented. The main result is the existence of a decreasing sequence of temperatures for which the Gibbs states have arbitrarily long periods. It is possible that the sequence accumulates at nonzero temperature, giving rise to a quasiperiodic equilibrium state.

Statistical mechanics of spatial evolutionary games

We discuss the long-run behaviour of stochastic dynamics of many interacting players in spatial evolutionary games. In particular, we investigate the effect of the number of players and the noise level on the stochastic stability of Nash equilibria. We discuss similarities and differences between systems of interacting players maximizing their individual payoffs and particles minimizing their interaction energy. We use concepts and techniques of statistical mechanics to study game-theoretic models. In order to obtain results in the case of the so-called potential games, we analyse the thermodynamic limit of the appropriate models of interacting particles.

Stochastic stability in spatial games

We discuss similarities and differences between systems of interacting players maximizing their individual payoffs and particles minimizing their interaction energy. Long-run behavior of stochastic dynamics of spatial games with multiple Nash equilibria is analyzed. In particular, we construct an example of a spatial game with three strategies, where stochastic stability of Nash equilibria depends on the number of players and the kind of dynamics.

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.

The unstable chemical structure of quasicrystalline alloys

Abstract We show that a typical toy model of an ordered quasicrystalline alloy has a range of stoichiometries.

How low temperature causes long-range order

The author shows that for almost all interactions, that is, generically, ground states of classical lattice gas models must have either long-range order or be completely uniform.

Devil's staircase for nonconvex interactions

We study rigorously ground-state orderings of particles in one-dimensional classical lattice gases with nonconvex interactions. Such systems serve as models of adsorption on crystal surfaces. In the considered models, the energy of adsorbed particles is a sum of two components, each one representing the energy of a one-dimensional lattice gas with two-body interactions in one of the two orthogonal lattice directions. This feature reduces two-dimensional problems to one-dimensional ones. The interaction energy in each direction is assumed here to be repulsive and strictly convex only from distance 2 on, while its value at distance 1 can be positive or negative, but close to zero. We show that if the decay rate of the interactions is fast enough, then particles form 2-particle lattice-connected aggregates (dimers) which are distributed in the same most homogeneous way as particles whose interaction is strictly convex everywhere. Moreover, despite the lack of convexity, the density of particles vs. the chemical potential appears to be a fractal curve known as the complete devils staircase.

Frustration without competing interactions

The concept of frustration is investigated following an idea of Anderson. A simple, frustrated in Andersons sense, nonrandom classical lattice spin system without competing interactions is discussed, which exhibits infinitely many equilibrium states at low temperature. The overlap distribution function is calculated exactly to be a delta function at zero.