Curie-Weiss Model Research Articles

The convex distance inequality for dependent random variables, with applications to the stochastic travelling salesman and other problems

We prove concentration inequalities for general functions of weakly dependent random variables satisfying the Dobrushin condition. In particular, we show Talagrand's convex distance inequality for this type of dependence. We apply our bounds to a version of the stochastic salesman problem, the Steiner tree problem, the total magnetisation of the Curie-Weiss model with external field, and exponential random graph models. Our proof uses the exchangeable pair method for proving concentration inequalities introduced by Chatterjee (2005). Another key ingredient of the proof is a subclass of $(a,b)$-self-bounding functions, introduced by Boucheron, Lugosi and Massart (2009).

Curie–Weiss magnet—a simple model of phase transition

The Curie–Weiss model is an exactly solvable model of ferromagnetism that allows one to study thermodynamic functions in detail, in particular their properties near the critical temperature. In this model every magnetic moment interacts with every other magnetic moment. Because of its simplicity and the correctness of at least some of its predictions, the Curie–Weiss model occupies an important place in the statistical mechanics literature and its application to information theory. It is frequently presented as an introduction to the Ising model or to spin-glass models, and usually only general features of the Curie–Weiss model are presented. In this paper, we discuss the properties of this model in a rather detailed way. We present the exact, approximate and numerical results for this particular model. The exact expression for the limiting magnetic field is derived.

Information geometry, phase transitions, and the Widom line: Magnetic and liquid systems

We study information geometry of the thermodynamics of first and second order phase transitions, and beyond criticality, in magnetic and liquid systems. We establish a universal microscopic characterization of such phase transitions via a conjectured equality of the correlation lengths ξ in co-existing phases, where ξ is related to the scalar curvature of the equilibrium thermodynamic state space. The 1-D Ising model, and the mean-field Curie–Weiss model are discussed, and we show that information geometry correctly describes the phase behavior for the latter. The Widom lines for these systems are also established. We further study a toy model for the thermodynamics of liquid–liquid phase co-existence, and show that our method provides a simple and direct way to obtain its phase behavior and the location of the Widom line. Our analysis points towards the possibility of multiple Widom lines in liquid systems.

Mean field spin glasses treated with PDE techniques

Following an original idea of Guerra, in these notes we analyze the Sherrington-Kirkpatrick model from different perspectives, all sharing the underlying approach which consists in linking the resolution of the statistical mechanics of the model (e.g. solving for the free energy) to well-known partial differential equation (PDE) problems (in suitable spaces). The plan is then to solve the related PDE using techniques involved in their native field and lastly bringing back the solution in the proper statistical mechanics framework. Within this strand, after a streamlined test-case on the Curie-Weiss model to highlight the methods more than the physics behind, we solve the SK both at the replica symmetric and at the 1–RSB level, obtaining the correct expression for the free energy via an analogy to a Fourier equation and for the self-consistencies with an analogy to a Burger equation, whose shock wave develops exactly at critical noise level (triggering the phase transition). Our approach, beyond acting as a new alternative method (with respect to the standard routes) for tackling the complexity of spin glasses, links symmetries in PDE theory with constraints in statistical mechanics and, as a novel result from the theoretical physics perspective, we obtain a new class of polynomial identities (namely of Aizenman-Contucci type, but merged within the Guerra’s broken replica measures), whose interest lies in understanding, via the recent Panchenko breakthroughs, how to force the overlap organization to the ultrametric tree predicted by Parisi.

Stein’s method in high dimensions with applications

Let $h$ be a three times partially differentiable function on $R^n$, let $X=(X_1,\dots,X_n)$ be a collection of real-valued random variables and let $Z=(Z_1,\dots,Z_n)$ be a multivariate Gaussian vector. In this article, we develop Stein's method to give error bounds on the difference $E h(X) - E h(Z)$ in cases where the coordinates of $X$ are not necessarily independent, focusing on the high dimensional case $n\to\infty$. In order to express the dependency structure we use Stein couplings, which allows for a broad range of applications, such as classic occupancy, local dependence, Curie-Weiss model etc. We will also give applications to the Sherrington-Kirkpatrick model and last passage percolation on thin rectangles.

Large deviations principle for Curie–Weiss models with random fields

In this paper, we consider an extension of the classical Curie–Weiss model in which the global and deterministic external magnetic field is replaced by local and random external fields which interact with each spin of the system. We prove a large deviations principle for the so-called magnetization per spin Sn/n with respect to the associated Gibbs measure, where Sn/n is the scaled partial sum of spins. In particular, we obtain an explicit expression for the rate function, which enables an extensive study of the phase diagram in some examples. It is worth mentioning that the model considered in this paper covers, in particularly, both the case of i.i.d. random external fields (also known under the name of random field Curie–Weiss models) and the case of dependent random external fields generated by e.g. Markov chains or dynamical systems.

Moderate deviations for a Curie–Weiss model with dynamical external field

In the present paper we prove moderate deviations for a Curie–Weiss model with external magnetic field generated by a dynamical system, as introduced by Dombry and Guillotin-Plantard in [C. Dombry and N. Guillotin-Plantard, Markov Process. Related Fields 15 (2009) 1–30]. The results extend those already obtained for the Curie–Weiss model without external field by Eichelsbacher and Lowe in [P. Eichelsbacher and M. Lowe, Markov Process. Related Fields 10 (2004) 345–366]. The Curie–Weiss model with dynamical external field is related to the so called dynamic ℤ-random walks (see [N. Guillotin-Plantard and R. Schott, Theory and applications, Elsevier B. V., Amsterdam (2006).]). We also prove a moderate deviation result for the dynamic ℤ-random walk, completing the list of limit theorems for this object.

From Stein identities to moderate deviations

Stein's method is applied to obtain a general Cramer-type moderate deviation result for dependent random variables whose dependence is defined in terms of a Stein identity. A corollary for zero-bias coupling is deduced. The result is also applied to a combinatorial central limit theorem, a general system of binary codes, the anti-voter model on a complete graph, and the Curie-Weiss model. A general moderate deviation result for independent random variables is also proved.

Variational Description of Gibbs-non-Gibbs Dynamical Transitions for the Curie-Weiss Model

We perform a detailed study of Gibbs-non-Gibbs transitions for the Curie-Weiss model subject to independent spin-flip dynamics (“infinite-temperature” dynamics). We show that, in this setup, the program outlined in van Enter et al. (Moscow Math J 10:687–711, 2010) can be fully completed, namely, Gibbs-non-Gibbs transitions are equivalent to bifurcations in the set of global minima of the large-deviation rate function for the trajectories of the magnetization conditioned on their endpoint. As a consequence, we show that the time-evolved model is non-Gibbs if and only if this set is not a singleton for some value of the final magnetization. A detailed description of the possible scenarios of bifurcation is given, leading to a full characterization of passages from Gibbs to non-Gibbs and vice versa with sharp transition times (under the dynamics, Gibbsianness can be lost and can be recovered). Our analysis expands the work of Ermolaev and Külske (J Stat Phys 141:727–756, 2010), who considered zero magnetic field and finite-temperature spin-flip dynamics. We consider both zero and non-zero magnetic field, but restricted to infinite-temperature spin-flip dynamics. Our results reveal an interesting dependence on the interaction parameters, including the presence of forbidden regions for the optimal trajectories and the possible occurrence of overshoots and undershoots in time of the initial magnetization of the optimal trajectories. The numerical plots provided are obtained with the help of MATHEMATICA.

Moderate Deviations for Random Field Curie-Weiss Models

The random field Curie-Weiss model is derived from the classical Curie-Weiss model by replacing the deterministic global magnetic field by random local magnetic fields. This opens up a new and interestingly rich phase structure. In this setting, we derive moderate deviations principles for the random total magnetization $S_n$, which is the partial sum of (dependent) spins. A typical result is that under appropriate assumptions on the distribution of the local external fields there exist a real number $m$, a positive real number $\lambda$, and a positive integer $k$ such that $(S_n-nm)/n^{\alpha}$ satisfies a moderate deviations principle with speed $n^{1-2k(1-\alpha)}$ and rate function $\lambda x^{2k}/(2k)!$, where $1-1/(2(2k-1)) < \alpha < 1$.

BK-type inequalities and generalized random-cluster representations

Recently, van den Berg and Jonasson gave the first substantial extension of the BK inequality for non-product measures: they proved that, for $$k$$ -out-of- $$n$$ measures, the probability that two increasing events occur disjointly is at most the product of the two individual probabilities. We show several other extensions and modifications of the BK inequality. In particular, we prove that the antiferromagnetic Ising Curie–Weiss model satisfies the BK inequality for all increasing events. We prove that this also holds for the Curie–Weiss model with three-body interactions under the so-called negative lattice condition. For the ferromagnetic Ising model we show that the probability that two events occur ‘cluster-disjointly’ is at most the product of the two individual probabilities, and we give a more abstract form of this result for arbitrary Gibbs measures. The above cases are derived from a general abstract theorem whose proof is based on an extension of the Fortuin–Kasteleyn random-cluster representation for all probability distributions and on a ‘folding procedure’ which generalizes an argument of Reimer.

Fluctuation relations for equilibrium states with broken discrete symmetries

Relationships are obtained expressing the breaking of spin-reversal symmetry by an external magnetic field in Gibbsian canonical equilibrium states of spin systems under specific assumptions. These relationships include an exact fluctuation relation for the probability distribution of the magnetization, as well as a relation between the standard thermodynamic entropy, an associated spin-reversed entropy or coentropy, and the product of the average magnetization with the external field, as a non-negative Kullback–Leibler divergence. These symmetry relations are applied to the model of noninteracting spins, the 1D and 2D Ising models, and the Curie–Weiss model, all in an external magnetic field. The results are drawn by analogy with similar relations obtained in the context of nonequilibrium physics.

A neural computation model for decision-making times

We introduce two new models for decision-making times for a two-choice decision task with no a priori bias. One of the models is the mean-field Curie–Weiss model of neural computation, and the other is based on dynamics near an unstable equilibrium under a small noise perturbation. As in the existing literature, we interpret exit times as reaction times and show that our models lead to a specific shape of the exit time distributions in the vanishing noise limit. We test the distribution shape against experimental data and show that for almost 90% of the participants, reaction times are described well by the model. Among the features of our model are: the dependence of the exit distribution only on two parameters, the elegance of rigorous mathematical analysis, and the microscopic nature of the noise.

On quantum mean-field models and their quantum annealing

This paper deals with fully connected mean-field models of quantum spins withp-body ferromagnetic interactions and a transverse field. Forp = 2 this corresponds to the quantum Curie–Weiss model (a special case of theLipkin–Meshkov–Glick model) which exhibits a second-order phase transition, while forp > 2 the transition is first order. We provide a refined analytical description both of the staticand of the dynamic properties of these models. In particular we obtain analytically theexponential rate of decay of the gap at the first-order transition. We also study the slowannealing from the pure transverse field to the pure ferromagnet (and vice versa) anddiscuss the effect of the first-order transition and of the spinodal limit of metastabilityon the residual excitation energy, both for finite and exponentially divergentannealing times. In the quantum computation perspective this quantity wouldassess the efficiency of the quantum adiabatic procedure as an approximationalgorithm.

Refining the Spin Hamiltonian in the Spin-12Kagome Lattice AntiferromagnetZnCu3(OH)6Cl2Using Single Crystals

We report thermodynamic measurements of the S=1/2 kagome lattice antiferromagnet ZnCu3(OH)6Cl2, a promising candidate system with a spin-liquid ground state. Using single crystal samples, the magnetic susceptibility both perpendicular and parallel to the kagome plane has been measured. A small, temperature-dependent anisotropy has been observed, where χ(z)/χ(p)>1 at high temperatures and χ(z)/χ(p)<1 at low temperatures. Fits of the high-temperature data to a Curie-Weiss model also reveal an anisotropy. By comparing with theoretical calculations, the presence of a small easy-axis exchange anisotropy can be deduced as the primary perturbation to the dominant Heisenberg nearest neighbor interaction. These results have great bearing on the interpretation of theoretical calculations based on the kagome Heisenberg antiferromagnet model to the experiments on ZnCu3(OH)6Cl2.

Chains of mean-field models

We consider a collection of Curie–Weiss (CW) spin systems, possibly with a random field,each of which is placed along the positions of a one-dimensional chain. The CW systemsare coupled together by a Kac-type interaction in the longitudinal direction of the chainand by an infinite-range interaction in the direction transverse to the chain. Ourmotivations for studying this model come from recent findings in the theory oferror-correcting codes based on spatially coupled graphs. We find that, althoughmuch simpler than the codes, the model studied here already displays similarbehavior. We are interested in the van der Waals curve in a regime where the sizeof each Curie–Weiss model tends to infinity, and the length of the chain andrange of the Kac interaction are large but finite. Below the critical temperature,and with appropriate boundary conditions, there appears a series of equilibriumstates representing kink-like interfaces between the two equilibrium states of theindividual system. The van der Waals curve oscillates periodically around theMaxwell plateau. These oscillations have a period inversely proportional to the chainlength and an amplitude exponentially small in the range of the interaction; inother words, the spinodal points of the chain model lie exponentially close to thephase transition threshold. The amplitude of the oscillations is closely related to aPeierls–Nabarro free energy barrier for the motion of the kink along the chain. Analogiesto similar phenomena and their possible algorithmic significance for graphicalmodels of interest in coding theory and theoretical computer science are pointedout.

Magnetism of a Nitrogen-Implanted TiO2 Single Crystal

Single rutile crystal TiO2 was implanted using nitrogen ions with energy of 60 keV. The microstructure, ultraviolet-visible light absorption spectra, conductivity and magnetism are investigated. Except for the nitrogen dopant, no impurity can be detected by x-ray diffraction and x-ray photoelectron spectra. The absorption in the visible light region is enhanced with nitrogen implantation dose increasing. By measuring the temperature dependence of resistance, it is found that the sample implanted with 1 × 1018 ions/cm2 is changes from insulating to semiconducting, and the variable range hopping is the main conducting mechanism. Room-temperature ferromagnetism is also obtained in this sample. The magnetism as a function of temperature can be well fitted using the three-dimensional spin wave model plus the Curie-Weiss model, indicating that there is a mixed phase of ferromagnetism and paramagnetism.

Scaling Limits for Multi-species Statistical Mechanics Mean-Field Models

We study the limiting thermodynamic behavior of the normalized sums of spins in multi-species Curie-Weiss models. We find sufficient conditions for the limiting random vector to be Gaussian (or to have an exponential distribution of higher order) and compute the covariance matrix in terms of model parameters.

Nonnormal approximation by Stein’s method of exchangeable pairs with application to the Curie–Weiss model

Let $(W,W')$ be an exchangeable pair. Assume that \[E(W-W'|W)=g(W)+r(W),\] where $g(W)$ is a dominated term and $r(W)$ is negligible. Let $G(t)=\int_0^tg(s)\,ds$ and define $p(t)=c_1e^{-c_0G(t)}$, where $c_0$ is a properly chosen constant and $c_1=1/\int_{-\infty}^{\infty}e^{-c_0G(t)}\,dt$. Let $Y$ be a random variable with the probability density function $p$. It is proved that $W$ converges to $Y$ in distribution when the conditional second moment of $(W-W')$ given $W$ satisfies a law of large numbers. A Berry-Esseen type bound is also given. We use this technique to obtain a Berry-Esseen error bound of order $1/\sqrt{n}$ in the noncentral limit theorem for the magnetization in the Curie-Weiss ferromagnet at the critical temperature. Exponential approximation with application to the spectrum of the Bernoulli-Laplace Markov chain is also discussed.

Notes on ferromagnetic diluted p-spin model

In this paper we develop the interpolating cavity field technique for the mean field ferromagnetic p-spin. The model we introduce is a natural extension of the diluted Curie–Weiss model to p > 2 spin interactions. Several properties of the free energy are analyzed and, in particular, we show that it recovers the expressions already known for p = 2 models and for p > 2 fully connected models. Further, as the model lacks criticality, we present extensive numerical simulations to evidence the presence of a first-order phase transition and deepen the behaviour at the transition line. Overall, a good agreement is obtained among analytical results, numerics and previous works.