Mean Field Dilute Ferromagnet: High Temperature and Zero Temperature Behavior

Sanctis, Luca de; Guerra, Francesco

doi:10.1007/s10955-008-9575-2

Cited by 28 publications

(48 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Besides regular lattices, in recent years much attention has been devoted to the setting in which the spin variables are placed on the vertices of random graphs [1,10,11,12,13,14,15,16,17,23,25]. Such random graphs aim to model emergent properties of complex systems consisting of many interacting agents described by a network.…”

Section: Ising Models On Random Graphsmentioning

confidence: 99%

Annealed central limit theorems for the Ising model on random graphs

Giardinà¹,

Giberti²,

Hofstad³

et al. 2016

ALEA

View full text Add to dashboard Cite

The aim of this paper is to prove central limit theorems with respect to the annealed measure for the magnetization rescaled by √ N of Ising models on random graphs. More precisely, we consider the general rank-1 inhomogeneous random graph (or generalized random graph), the 2-regular configuration model and the configuration model with degrees 1 and 2. For the generalized random graph, we first show the existence of a finite annealed inverse critical temperature 0 ≤ β an c < ∞ and then prove our results in the uniqueness regime, i.e., the values of inverse temperature β and external magnetic field B for which either β < β an c and B = 0, or β > 0 and B = 0. In the case of the configuration model, the central limit theorem holds in the whole region of the parameters β and B, because phase transitions do not exist for these systems as they are closely related to one-dimensional Ising models. Our proofs are based on explicit computations that are possible since the Ising model on the generalized random graph in the annealed setting is reduced to an inhomogeneous Curie-Weiss model, while the analysis of the configuration model with degrees only taking values 1 and 2 relies on that of the classical one-dimensional Ising model.

show abstract

Section: Ising Models On Random Graphsmentioning

confidence: 99%

Annealed central limit theorems for the Ising model on random graphs

Giardinà¹,

Giberti²,

Hofstad³

et al. 2016

ALEA

View full text Add to dashboard Cite

show abstract

“…As the theory is no longer Gaussian, we need infinite sets of random fields (mapping the presence of multi-overlaps in standard dilution [3] [43] and no longer only the first two momenta of the distributions). Of course we recover the proper free energy by evaluating the trial A(t) at t = 1, (A(β, α, a) = A(t = 1)), which we want to obtain by using the fundamental theorem of calculus:…”

Section: Free Energy Trough Extended Double Stochastic Stabilitymentioning

confidence: 99%

Equilibrium statistical mechanics on correlated random graphs

Barra¹,

Agliari²

2011

J. Stat. Mech.

View full text Add to dashboard Cite

Biological and social networks have recently attracted enormous attention between physicists. Among several, two main aspects may be stressed: A non trivial topology of the graph describing the mutual interactions between agents exists and/or, typically, such interactions are essentially (weighted) imitative. Despite such aspects are widely accepted and empirically confirmed, the schemes currently exploited in order to generate the expected topology are based on a-priori assumptions and in most cases still implement constant intensities for links. Here we propose a simple shift [−1, +1] → [0, +1] in the definition of patterns in an Hopfield model to convert frustration into dilution: By varying the bias of the pattern distribution, the network topology -which is generated by the reciprocal affinities among agents (the Hebbian kernel)-crosses various well known regimes (fully connected, linearly diverging connectivity, extreme dilution scenario, no network), coupled with small world properties, which, in this context, are emergent and no longer imposed a-priori. The model is investigated at first focusing on these topological properties of the emergent network, then its thermodynamics is analytically solved (at a replica symmetric level) by extending the double stochastic stability technique, and presented together with its fluctuation theory for a picture of criticality: both a statistical mechanics and a topological phase diagrams are obtained. Overall the picture depicted from statistical mechanics is quite intuitive: at least at equilibrium, dilution (of whatever kind) simply decreases the strength of the coupling felt by the spins, but leaves the paramagnetic/ferromagnetic flavors unchanged. The main difference with respect to previous investigations and a naive picture is that within our approach replicas do not appear: instead of (multi)-overlaps as order parameters, we introduce a class of magnetizations on all the possible sub-graphs belonging to the main one investigated: As a consequence, for these objects a closure for a self-consistent relation is achieved. 1As we will see, small values of a give rise to highly correlated, diluted networks, while, as a gets larger the network gets more and more connected and correlation among links vanishes.Even though the theory is defined at each finite V and L, as standard in statistical mechanics, we are interested in the large V behavior (such that, under central limit theorem permissions, deviations from averaged values become negligible and the theory predictive). To this task we find meaningful to let even L diverge linearly with the system size (to bridge conceptually to high storage neural networks), such that lim V →∞ L/V = α defines α as another control parameter. Finally, since we are interested in the regime of large V and large L we will often confuse V with V − 1 and L with L − 1. 4

show abstract

“…The study of diluted ferromagnets [1][2][3][4][5][6] dates back to several years ago, following two main paths sometimes overlapping, i.e. the statistical mechanics approach to lattices, and the graph theory approach to networks [7,8].…”

Section: Introductionmentioning

confidence: 99%

Dilution of Ferromagnets via a Random Graph‐Based Strategy

Javarone

Marinazzo

2018

Complexity

View full text Add to dashboard Cite

The dynamics and behavior of ferromagnets have a great relevance even beyond the domain of statistical physics. In this work, we propose a Monte Carlo method, based on random graphs, for modeling their dilution. In particular, we focus on ferromagnets with dimension D ≥ 4, which can be approximated by the Curie-Weiss model. Since the latter has as graphic counterpart a complete graph, a dilution can be in this case viewed as a pruning process. Hence, in order to exploit this mapping, the proposed strategy uses a modified version of the Erdős-Renyi graph model. In doing so, we are able both to simulate a continuous dilution, and to realize diluted ferromagnets in one step. The proposed strategy is studied by means of numerical simulations, aimed to analyze main properties and equilibria of the resulting diluted ferromagnets. To conclude, we also provide a brief description of further applications of our strategy in the field of complex networks. *

show abstract

Mean Field Dilute Ferromagnet: High Temperature and Zero Temperature Behavior

Cited by 28 publications

References 23 publications

Annealed central limit theorems for the Ising model on random graphs

Annealed central limit theorems for the Ising model on random graphs

Equilibrium statistical mechanics on correlated random graphs

Dilution of Ferromagnets via a Random Graph‐Based Strategy

Contact Info

Product

Resources

About