van Aernout Enter

Possible loss and recovery of Gibbsianness during the stochastic evolution of Gibbs measures

Abstract: We consider Ising-spin systems starting from an initial Gibbs measure ν and evolving under a spin-flip dynamics towards a reversible Gibbs measure μ≠ν. Both ν and μ are assumed to have a translation-invariant finite-range interaction. We study the Gibbsian character of the measure νS(t) at time t and show the following:(1) For all ν and μ, νS(t) is Gibbs for small t.(2) If both ν and μ have a high or infinite temperature, then νS(t) is Gibbs for all t > 0.(3) If ν has a low non-zero temperature and a zero magnetic field and μ has a high or infinite temperature, then νS(t) is Gibbs for small t and non-Gibbs for large t.(4) If ν has a low non-zero temperature and a non-zero magnetic field and μ has a high or infinite temperature, then νS(t) is Gibbs for small t, non-Gibbs for intermediate t, and Gibbs for large t.The regime where μ has a low or zero temperature and t is not small remains open. This regime presumably allows for many different scenarios.

Finite-Size Effects for Some Bootstrap Percolation Models

The consequences of Schonmanns new proof that the critical threshold is unity for certain bootstrap percolation models are explored. It is shown that this proof provides an upper bound for the finite-size scaling in these systems. Comparison with data for one case demonstrates that this scaling appears to give the correct asymptotics. We show that the threshold for a finite system of sizeL scales asO[ln(lnL)] for the isotropic model in three dimensions where sites that fail to have at least four neighbors are culled.

Chaotic temperature dependence at zero temperature.

We present a class of examples of nearest-neighbour, bounded-spin models, in which the low-temperature Gibbs measures do not converge as the temperature is lowered to zero, in any dimension.

Differentiability properties of the pressure in lattice systems

In two recent papers Ruelle gave a heuristic theory of phase transitions, using some techniques introduced by Israel. He proves a version of Gibbs phase rule, assuming a differentiability condition for the pressure. Ruelle already pointed out that his condition cannot always hold. In this paper we prove that the interaction spaces which he considers are in general too large for his condition to hold. We also show that the version of the Gibbs phase rule which is a consequence of this condition does not hold in general. Moreover we give some constraints on the analyticity properties of the pressure.

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.

The Order Parameter in a Spin Glass

Various possible precise definitions of an Edwards-Anderson type of order parameter for an Ising model spin glass are considered, using boundary conditions for a finite system, states of an infinite system, and a duplicate-system approach. Several of these definitions are shown to yield identical results.

A Note on the Stability of Phase Diagrams in Lattice Systems

We construct a class of non-symmetry breaking pair interactions, which change the phase diagram of then.n. Ising and classicalX Y model. Furthermore we improve earlier obtained constraints on the decrease of interactions, necessary to get analyticity properties of the pressure in manifolds of non-symmetry breaking interactions.

A Remark on Different Norms and Analyticity for Many-Particle Interactions

We compare a recent result of Dobrushin and Martirosyan with previous results by Gallavotti and Miracle-Sole and by Israel and point out that the analytic behavior at high temperatures for many-particle interactions is different depending on whether the interactions are weighted with a lattice-gas or Ising norm or, on the other hand, with the supremum norm.

Infinitely many states and stochastic symmetry in a Gaussian Potts-Hopfield model

We study a Gaussian Potts-Hopfield model. Whereas for Ising spins and two disorder variables per site the chaotic pair scenario is realized, we find that for q-state Potts spins q (q - 1)-tuples occur. Beyond the breaking of a continuous stochastic symmetry, we study the fluctuations and obtain the Newman-Stein metastate description for our model.