Constrained variational problem with applications to the Ising model

We continue our study of the behavior of the two-dimensional nearest neighbor ferromagnetic Ising model under an external magnetic fieldh, initiated in our earlier work. We strengthen further a result previously proven by Martirosyan at low enough temperature, which roughly states that for finite systems with (–)-boundary conditions under a positive external field, the boundary effect dominates in the system 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 system. In our earlier work this result was extended to every subcritical value of the temperature. Here for every subcritical value of the temperature we show the existence of a critical valueB ₀ (T) which separates the two regimes specified above. We also find the asymptotic shape of the region occupied by the (+)-phase in the second regime, which turns out to be a squeezed Wulff shape. The main step in our study is the solution of the variational problem of finding the curve minimizing the Wulff functional, which curve is constrained to the unit square. Other tools used are the results and techniques developed to study large deviations for the block magnetization in the absence of the magnetic field, extended to all temperatures below the critical one.     

Surface order large deviations of phase interfaces for the continuum Widom-Rowlinson model in high density limit

We establish a surface order large deviation principle characterising, in the phase coexistence region, the exponential decay rates for the probabilities of macroscopic fluctuations of phase-separating interfaces for the continuum Widom-Rowlinson binary gas, with the thermodynamic and high fugacity limits taken simultaneously. The large deviation rate function is given by an isotropic surface energy functional and hence it attains its minimum for balls which are the most favourable shapes of ‘droplets’ of dominated phase within the ‘ocean’ of dominating phase.     

Metastability in the two-dimensional ising model

Metastability in the Ising model is studied in two ways. In a dynamical Monte Carlo model, metastable magnetization and lifetime are measured for various magnetic fields and low temperatures. Following up a proposed relation between analytic continuation of transfer matrix eigenvalues and metastability, transfer matrix eigenvalues are studied. We examine the extent to which these approaches agree. The Monte Carlo data also provide quantitative support for the critical droplet model for decay.     

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.     

A microscopic justification of the Wulff construction

We report results about a rigorous microscopic justification of the Wulff construction for the two-dimensional Ising model at low temperatures and under periodic boundary conditions. The idea of the proof is sketched.     

Large deviations for noninteracting infinite-particle systems

A large deviation property is established for noninteracting infinite particle systems. Previous large deviation results obtained by the authors involved a singleI-function because the cases treated always involved a unique invariant measure for the process. In the context of this paper there is an infinite family of invariant measures and a corresponding infinite family ofI-functions governing the large deviations.     

Large deviations estimates for the multiscale analysis of heart rate variability

In the realm of multiscale signal analysis, multifractal analysis provides a natural and rich framework to measure the roughness of a time series. As such, it has drawn special attention of both mathematicians and practitioners, and led them to characterize relevant physiological factors impacting the heart rate variability. Notwithstanding these considerable progresses, multifractal analysis almost exclusively developed around the concept of Legendre singularity spectrum, for which efficient and elaborate estimators exist, but which are structurally blind to subtle features like non-concavity or, to a certain extent, non scaling of the distributions. Large deviations theory allows bypassing these limitations but it is only very recently that performing estimators were proposed to reliably compute the corresponding large deviations singularity spectrum. In this article, we illustrate the relevance of this approach, on both theoretical objects and on human heart rate signals from the Physionet public database. As conjectured, we verify that large deviations principles reveal significant information that otherwise remains hidden with classical approaches, and which can be reminiscent of some physiological characteristics. In particular we quantify the presence/absence of scale invariance of RR signals.     

Phase diagram of the half-infinite ising model

The phase diagram is analyzed rigorously, and in particular the wetting transition is discussed.     

Exact expressions for row correlation functions in the isotropicd=2 ising model

We present exact explicit expressions for the row spin-spin correlation functions _{00 n}0 in the isotropicd= 2 Ising model, in terms of elliptic integrals, forn 5. We also give a general structural formula for _{00 n}0.     

Large deviations for the stochastic derivative Ginzburg-Landau equation with multiplicative noise

This paper first proves the existence of a unique mild solution to the stochastic derivative Ginzburg-Landau equation. The fixed point theorem for the corresponding truncated equation is used as the main tool. Since we restrict our study to the one-dimensional case, it is not necessary to introduce another Banach space and thus the estimates of the stochastic convolutions in the Banach space are avoided. Secondly, we also consider large deviations for the stochastic derivative Ginzburg-Landau equation perturbed by a small noise. Since the underlying space considered is Polish, using the weak convergence approach, we establish a large deviations principle by proving a Laplace principle.     

Two- and three-spin triangular ising model: Variational approximations

The implications of the known hard-hexagon lattice gas results for the triangular Ising model with both pair and triplet interactions are pointed out. Employing an appropriate generalization of the variational method of Baxter we determine, using the lowest-order approximation, the phase boundaries for this model when the pair interactions are ferromagnetic. Higher approximations are presented for the case of pure triplet interactions and the resulting phase diagrams are in excellent agreement with all exactly known results.     

Transition probabilities and stochastic equations for the mean field ising model

The microscopic transition rate is briefly calculated from quantum principles to derive the microscopic master equation. By introducing _p, the phenomenological time, and coarse graining W_p, the transition rate, a complete normalized phenomenological transition rate is obtained. The Langer form is then approximately obtained.Supported in part by the Robert A. Welch Foundation.On leave of absence from the Institute of Theoretical Physics, Academia Sinica, Beijing, China.     

Computation of critical exponents for two-dimensional ising model on a cellular automaton

Static critical exponents for the two-dimensional Ising model are computed on a cellular automaton. The analysis of the data within the framework of the finite-size scaling theory reproduces their well-established values.     

The axial ising model for martensitic transformations and pseudoelasticity

A theory based on the Ising axial model for the martensitic transformation between close-packed structures in an external stress field is proposed. The quasi-spin Hamiltonian was derived for a stratified close-packed crystal which under certain limitations has the form of a Hamiltonian of the ANNNI-model wherein “the exchange integrals” are expressed in terms of the interlayer interaction potentials. The dependence of the strain on the shear stress is calculated for alloys showing the pseudoelasticity effect and it is shown that the number of steps in the deformation curve is defined by the radius of the interlayer interaction. The theory proposed is applied to the explanation of the stress-induced martensitic transformations in Cu-Al-Ni alloys.     

The Spectral Gap for the Kawasaki Dynamics at Low Temperature

In this paper we analyze the convergence to equilibrium of Kawasaki dynamics for the Ising model in the phase coexistence region. First we show, in strict analogy with the nonconservative case, that in any lattice dimension, for any boundary condition and any positive temperature and particle density, the spectral gap in a box of side L does not shrink faster than a negative exponential of the surface L ^d–1. Then we prove that, in two dimensions and for free boundary condition, the spectral gap in a box of side L is smaller than a negative exponential of L provided that the temperature is below the critical one and the particle density satisfies (*_–, *₊), where *_± represents the particle density of the plus and minus phase, respectively.     

Upper bounds on the critical temperature for various ising models

Upper bounds are obtained for spin ±1 systems. In the case of only nearestneighbor interactions on, for example, the square lattice we obtain _cJ>0.3592. The method's strength is seen when considering systems with longer-range interactions. For example, we obtain _cJ>0.360 compared to the previous best bound of _c J 0.345 for the one-dimensional lattice with 1/r ² interactions. The method relies upon an identity between correlation functions and then the use of correlation inequalities to obtain the final bounds.     

A lower bound for the memory capacity in the Potts-Hopfield model

We consider Potts-Hopfield networks of sizeN. We prove the result: _c >0 such that for all 0<< _c we can find, >0 in such a way that, whenN, we can store N patterns, all of them being sorrounded by -energy barriers at distance.     

二维伊辛模型相变临界现象的元胞自动机模拟

用元胞自动机模型模拟二维伊辛模型的相变临界现象,得出了相变图和时空演化图;当不存在外场时,数值模拟得到的临界温度与理论值精确吻合。