Roberto H. Schonmann

Critical droplets and metastability for a Glauber dynamics at very low temperatures

We consider the metastable behavior in the so-called pathwise approach of a ferromagnetic spin system with a Glauber dynamics in a finite two dimensional torus under a positive magnetic field in the limit as the temperature goes to zero. First we consider the evolution starting from a single rectangular droplet of spins +1 in a sea of spins −1. We show that small droplets are likely to disappear while large droplets are likely to grow; the threshold between the two cases being sharply defined and depending only on the external field. This result is used to prove that starting from the configuration with all spins down (−1) the pattern of evolution leading to the more stable configuration with all spins up (+1) approaches, as the temperature vanishes, a metastable behavior: the system stays close to −1 for an unpredictable time until a critical square droplet of a precise size is eventually formed and nucleates the decay to +1 in a relatively short time. The asymptotic magnitude of the total decay time is shown to be related to the height of an energy barrier, as expected from heuristic and mean field studies of metastability.

Behavior of droplets for a class of Glauber dynamics at very low temperature

SummaryWe consider a class of Glauber dynamics for the two-dimensional nearest neighbor ferromagnetic Ising model in which the rate with which each spin flips depends only on the increment in energy caused by its flip in a monotonic non-increasing fashion.We extend to this class results previously shown for the particular case of Metropolis dynamics [NS]. We show that for fixed volume and external field 0<h<1, at very low temperature small rectangular droplets of sping +1 in a sea of spins −1 tend to shrink, while large droplets tend to grow and cover the whole system. The threshold between the two behaviors is sharply defined, the critical length being 2/h.An example is given which shows that without the assumption of monotonicity of the rates this result may be false.We use the result on critical droplets to show that starting from the configuration with all spins down, the systems in the class that we consider evolve in a metastable fashion until the configuration with all spins up is reached.For similar systems in higher dimensions we show that under analogous conditions on the rates, small enough droplets are likely to shrink, while large enough droplets are likely to grow.

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.

Slow droplet-driven relaxation of stochastic Ising models in the vicinity of the phase coexistence region

We consider the stochastic Ising models (Glauber dynamics) corresponding to the infinite volume basic Ising model in arbitrary dimensiond≧2 with nearest neighbor interaction and under a positive external magnetic fieldh. Under minimal assumptions on the rates of flip (so that all the common choices are included), we obtain results which state that when the system is at low temperatureT, the relaxation time when the evolution is started with all the spins down blows up, whenh↘0, as exp(λ(T)/hd−1) (the precise results are lower and upper bounds of this form). Moreover, after a time which does not scale withh and before a time which also grows as an exponential of a multiple of 1/hd−1 ash↘0, the law of the state of the process stays, whenh is small, close to the minus-phase of the same Ising model without an external field. These results may be considered as a partial vindication of a conjecture raised by Aizenman and Lebowitz in connection to the metastable behavior of these stochastic Ising models.

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.

Projections of Gibbs measures may be non-Gibbsian

Consider the + phase of the two dimensional nearest neighbor ferromagnetic Ising model at a temperature belowTc. Let ν+ be the restriction of this measure to a coordinate axis. We prove that there is no one dimensional translation invariant summable interaction for which ν+ is a Gibbs measure. This is proven by showing that if such an interaction existed, ν+ would have large deviation properties different from those it actually has. Percolation methods are used in the proof.

Second order large deviation estimates for ferromagnetic systems in the phase coexistence region

AbstractWe consider thed-dimensional Ising model with ferromagnetic nearest neighbor interaction at inverse temperature β. Let

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

M_\Lambda = |\Lambda |^{ - 1} \sum\limits_{i \in \Lambda } {\sigma _i }