Projections of Gibbs measures may be non-Gibbsian

Consider the + phase of the two dimensional nearest neighbor ferromagnetic Ising model at a temperature belowT _c . Let ₊ be the restriction of this measure to a coordinate axis. We prove that there is no one dimensional translation invariant summable interaction for which ₊ is a Gibbs measure. This is proven by showing that if such an interaction existed, ₊ would have large deviation properties different from those it actually has. Percolation methods are used in the proof.Work supported by the U.S. Army Research Office through the Mathematical Sciences Institute at Cornell and by a NSF grant to Cornell. This work was finished while the author was visiting Rutgers University, being supported by the NSF grant 86-12369     

On the two-dimensional stochastic Ising model in the phase coexistence region near the critical point

We consider the two-dimensional stochastic Ising model in finite square with free boundary conditions, at inverse temperature >₀ and zero external field. Using duality and recent results of Ioffe on the Wulff construction close to the critical temperature, we extend some of the results obtained by Martinelli in the low-temperature regime to any temperature below the critical one. In particular we show that the gap in the spectrum of the generator of the dynamics goes to zero in the thermodynamic limit as an exponential of the side length of , with a rate constant determined by the surface tension along one of the coordinate axes. We also extend to the same range of temperatures the result due to Shlosman on the equilibrium large deviations of the magnetization with free boundary conditions.     

Exponential decay of connectivities in the two-dimensional ising model

We prove some results concerning the decay of connectivities in the low-temperature phase of the two-dimensional Ising model. These provide the bounds necessary to establish, nonperturbatively, large-deviation properties for block magnetizations in these systems. We also obtain estimates on the rate at which the finite-volume, plus-boundary-condition expectation of the spin at the origin converges to the spontaneous magnetization.On leave from São Paulo University, Brazil.     

Critical points of two-dimensional bootstrap percolation-like cellular automata

Cellular automata in two dimensions that generalize the bootstrap percolation dynamics are considered, focusing on the thresholdp _c of the initial density for convergence to total occupancy to occur; these models are classified according top _c being 0, 1, or strictly between these extreme values. Explicit upper and lower bounds are provided in the third case.     

On some growth models with a small parameter

Summary We consider the behavior of the asymptotic speed of growth and the asymptotic shape in some growth models, when a certain parameter becomes small. The basic example treated is the variant of Richardson's growth model on ^d in which each site which is not yet occupied becomes occupied at rate 1 if it has at least two occupied neighbors, at rate 1 if it has exactly 1 occupied neighbor and, of course, at rate 0 if it has no occupied neighbor. Occupied sites remain occupied forever. Starting from a single occupied site, this model has asymptotic speeds of growth in each direction (as time goes to infinity) and these speeds determine an asymptotic shape in the usual sense. It is proven that as tends to 0, the asymptotic speeds scale as ^1/d and the asymptotic shape, when renormalized by dividing it by ^1/d , converges to a cube. Other similar models which are partially oriented are also studied.The work of R.H.S. was supported by the N.S.F. through grant DMS 91-00725. In addition, both authors were supported by the Newton Institute in Cambridge. The authors thank the Newton Institute for its support and hospitality     

Complete analyticity for 2D Ising completed

We study the behavior of the two-dimensional nearest neighbor ferromagnetic Ising model under an external magnetic fieldh. We extend to every subcritical value of the temperature a result previously proven by Martirosyan at low enough temperature, and which roughly states that for finite systems with — boundary conditions under a positive external field, the boundary effect dominates in the bulk if the linear size of the system is of orderB/h withB small enough, while ifB is large enough, then the external field dominates in the bulk. As a consequence we are able to complete the proof that complete analyticity for nice sets holds for every value of the temperature and external field in the interior of the uniqueness region in the phase diagram of the model.The main tools used are the results and techniques developed to study large deviations for the block magnetization in the absence of the magnetic field, and recently extended to all temperatures below the critical one by Ioffe.     

Correlation lengths for oriented percolation

Oriented percolation has two correlation lengths, one in the space and one in the time direction. In this paper we define these quantities for the two-dimensional model in terms of the exponential decay of suitably chosen quantities, and study the relationship between the various definitions. The definitions are used in a companion paper to prove inequalities between critical exponents.     

For 2-D lattice spin systems weak mixing implies strong mixing

We prove that for finite range discrete spin systems on the two dimensional latticeZ ², the (weak) mixing condition which follows, for instance, from the Dobrushin-Shlosman uniqueness condition for the Gibbs state implies a stronger mixing property of the Gibbs state, similar to the Dobrushin-Shlosman complete analyticity condition, but restricted to all squares in the lattice, or, more generally, to all sets multiple of a large enough square. The key observation leading to the proof is that a change in the boundary conditions cannot propagate either in the bulk, because of the weak mixing condition, or along the boundary because it is one dimensional. As a consequence we obtain for ferromagnetic Ising-type systems proofs that several nice properties hold arbitrarily close to the critical temperature; these properties include the existence of a convergent cluster expansion and uniform boundedness of the logarithmic Sobolev constant and rapid convergence to equilibrium of the associated Glauber dynamics on nice subsets ofZ ², including the full lattice.Work partially supported by grant SC1-CT91-0695 of the Commission of European Communities and by grant DMS 91-00725 of the American NSF.     

The Triangle Condition for Contact Processes on Homogeneous Trees

We complete work of C. C. Wu, by showing that for contact processes on homogeneous trees with degree at least 3 the triangle condition is satisfied below the second critical point. In particular it holds at the first critical point and therefore at this critical point the contact process has mean-field critical exponents.     

The asymmetric contact process

We study a generalization of the Harris one-dimensional contact process in which the rates of infection to the right and left may be different.