Random field Ising model

Generalizing the Affine Framework to HJM and Random Field Models

We develop multispectral random field image models for use in image processing applications. The simultaneous autoregressive and Markov random field (MRF) models have been widely used in modeling intensity images. In this work we extend these models to include the more general multispectral case where images are represented by multiple intensity planes. In particular, we focus on the obvious application to color texture modeling using the RGB color model. For each model type we present the model equations, develop methods for synthesizing images based on these models and procedures for estimating the model parameters. In addition, the conditions necessary to ensure model validity are identified. We also provide experimental results which, substantiate the validity of these results. Color images synthesized from these models are shown to have the statistical characteristics implied by the model equations and parameters estimated from these images are very close to the known values from which the images were generated. In further experiments color random field models were fitted to natural texture samples. Images synthesized from these models are observed to be visually similar to the original images.

Modeling of non-stationary random field of undrained shear strength of soil for slope reliability analysis

I will give in this lecture an introductory overview of our present understanding of the Random Field Ising Model (RFIM). As this is a subject of a large literature and a long, turbulent history, I will restrict myself to a small number of topics, whose choice is very subjective. In particular, I will neither discuss experimental nor numerical work, except when it cannot be avoided. It is, however, important to realize why such a disproportionately large amount of attention has been devoted to this model. The first reason is that the RFIM is the simplest model which incorporates the dramatic effect of quenched-in disorder coupled linearly to an order-parameter. This effect has been seen in a steadily growing list of experiments (1). It was extensively studied for the first time in diluted antiferromagnets in a magnetic field (2) but the RFIM is now used to explain how impurities affect displacive transitions, how phase separation of binary fluids proceeds in porous media or gels, and how hydrogen dissolves in metallic alloys. Random fields also play an important role in charge-density wave systems.

Convex relaxation for IMSE optimal design in random-field models

For Abstract see ChemInform Abstract in Full Text.

We study a zero-temperature phase transition in the random field Ising model on scale-free networks with the degree exponent gamma. Using an analytic mean-field theory, we find that the spins are always in the ordered phase for gamma<3. On the other hand, the spins undergo a phase transition from an ordered phase to a disordered phase as the dispersion of the random fields increases for gamma>3. The phase transition may be either continuous or discontinuous depending on the shape of the random field distribution. We derive the condition for the nature of the phase transition. Numerical simulations are performed to confirm the results.

The random field Ising model with four spin interactions is studied within the finite cluster approximation (F.C.A), on square and simple cubic lattices. We have investigated the influence of the four spin interactions on the phase diagrams of the system. We have shown that for the simple cubic lattice, the tricritical behaviour due to the random field does not exist for every strength of the four spin interactions, and that he tricritical point due to four spin intenactions disappears when the value of the random field is large.

Bayesian hierarchical models usually model the risk surface on the same arbitrary geographical units for all data sources. Poisson/gamma random field models overcome this restriction as the underlying risk surface can be specified independently to the resolution of the data. Moreover, covariates may be considered as either excess or relative risk factors. We compare the performance of the Poisson/gamma random field model to the Markov random field (MRF)-based ecologic regression model and the Bayesian Detection of Clusters and Discontinuities (BDCD) model, in both a simulation study and a real data example. We find the BDCD model to have advantages in situations dominated by abruptly changing risk while the Poisson/gamma random field model convinces by its flexibility in the estimation of random field structures and by its flexibility incorporating covariates. The MRF-based ecologic regression model is inferior. WinBUGS code for Poisson/gamma random field models is provided.

The Random Field Ising Model (RFIM) represents one of the simplest models of cooperative behaviour with quenched disorder, and it is, in a way, complementary to the Ising Spin Glass which will be extensively treated later in this book. It accounts for the presence of a random external magnetic field which antagonizes the ordering induced by the ferromagnetic spin–spin interactions. From an experimental point of view, on the other hand, as shown by Fishman and Aharony (1979) and Cardy (1984), it is equivalent to a dilute anti-ferromagnet in a uniform field (see Belanger, 1998 for a recent review on experimental results).

We discuss some aspects of the problem of the equivalence of dilute antiferromagnets and random field Ising models. We first investigate for dilute antiferromagnets the validity of the arguments of Imry and Ma. It turns out that they are applicable, but some care is required concerning the role played by the so-called internal Peierls contours. Next we consider a hierarchical version of a dilute antiferromagnetic Ising model in the presence of a uniform magnetic field and show that a renormalization group transformation maps it exactly into a hierarchical version of the random field Ising model, thus proving their equivalence as far as the critical behavior is concerned. In particular this implies that phase transition with spontaneous magnetization occurs only for dimensiond>2. Finally we show that in the absence of internal Peierls contours both models, in their hierarchical versions, exhibit phase transition already in dimensiond=2.

Over the past ten years there has been a considerable amount of interest, both theoretical as well as experimental, in the effects of random fields on phase transitions, in particular, the Random Field Ising model (RFIM). Initially, the crucial issue seems to have been the lower critical dimensionality dc of RFIM1. In the literature arguments have been given in support of dc being equal to either 2 or 3 (see references 1, 2 and 3 for recent reviews of RFIM). However, recently it has been realized that, experimentally, RFIM systems get trapped in a nonegyilibrium state which relaxes very slowly, if at all, towards equilibrium4. In recent theoretical papers, Villain5 and Bruinsma and Aeppli6 have come to the conclusion that for d<5, the random field prevents the growth of an ordered state when the system is quenched to low temperatures from the paramagnetic regime.

We have performed measurements of the critical neutron scattering on CsCo0.83Mg0.17Br3, a dilute stacked triangular lattice (STL) Ising antiferromagnet (AF). A two component line shape associated with the critical fluctuations appears at a temperature coincident with T(N1) observed in pure CsCoBr3. Such scattering is indicative of fluctuations in prototypical random field Ising model (RFIM) systems. The random field domain state arises in this case due to geometrical frustration within the STL Ising AF, which gives rise to a three sublattice Néel state, in which one sublattice is disordered. Magnetic vacancies nucleate AF domains in which the vacancies reside on the disordered sublattice thereby generating a RFIM state in the absence of an applied magnetic field.

In this note we study the mean field equations for the $3d$ Random Field Ising Model. We discuss the phase diagram of the model, and we address the problem of finding if such equations admit more than one solution. We find two different critical values of $\beta$: one where the magnetization takes a non-zero expectation value, and one where we start to have more than one solution to the mean field equation. We find that, inside a given solution, there are no divergent correlation lengths.

We compare the ground state of the random-field Ising model with Gaussian distributed random fields, with its non-equilibrium hysteretic counterpart, the demagnetized state. This is a low energy state obtained by a sequence of slow magnetic field oscillations with decreasing amplitude. The main concern is how optimized the demagnetized state is with respect to the best-possible ground state. Exact results for the energy in d=1 show that in a paramagnet, with finite spin-spin correlations, there is a significant difference in the energies if the disorder is not so strong that the states are trivially almost alike. We use numerical simulations to better characterize the difference between the ground state and the demagnetized state. For d>=3 the random-field Ising model displays a disorder induced phase transition between a paramagnetic and a ferromagnetic state. The locations of the critical points R_c(DS), R_c(GS) differ for the demagnetized state and ground state. Consequently, it is in this regime that the optimization of the demagnetized stat is the worst whereas both deep in the paramagnetic regime and in the ferromagnetic one the states resemble each other to a great extent. We argue based on the numerics that in d=3 the scaling at the transition is the same in the demagnetized and ground states. This claim is corroborated by the exact solution of the model on the Bethe lattice, where the R_c's are also different.