Finite size effects for the Ising model on random graphs with varying dilution

Barré, Julien; Ciani, Antonia; Fanelli, Duccio; Bagnoli, Franco; Ruffo, Stefano

doi:10.1016/j.physa.2009.04.024

Cited by 7 publications

(8 citation statements)

References 16 publications

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“…The dilution coefficient γ may take any positive value and the number of bonds is about γN 2 . This model has been recently studied in a recent paper [5], while models with lesser dilution [7,8] ( number of bonds of order N ǫ with ǫ > 1 ) have been also considered.…”

Section: -Dilute Ferromagnetmentioning

confidence: 99%

Magnetization densities as replica parameters: The dilute ferromagnet

Serva

2010

Physica A: Statistical Mechanics and its Applications

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In this paper we compute exactly the ground state energy and entropy of the dilute ferromagnetic Ising model. The two thermodynamic quantities are also computed when a magnetic field with random locations is present. The result is reached in the replica approach frame by a class of replica order parameters introduced by Monasson [1]. The strategy is first illustrated considering the SK model, for which we will show the complete equivalence with the standard replica approach.Then, we apply to the diluted ferromagnetic Ising model with a random located magnetic field, which is mapped into a Potts model. This formalism can be, in principle, applied to all random systems, and we believe that it could be of help in many other contexts.

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Section: -Dilute Ferromagnetmentioning

confidence: 99%

Magnetization densities as replica parameters: The dilute ferromagnet

Serva

2010

Physica A: Statistical Mechanics and its Applications

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“…On the other hand, important aspects of rigidity percolation in a field have been addressed by C. Moukarzel [11]. Of course, other very interesting aspects that we do not treat for the moment are nucleation, metastability, frustration and dilution [7,12,13,14,15]. …”

Section: The Modelmentioning

confidence: 99%

Signature of Thermal Rigidity Percolation

Huerta

2013

J. Phys.: Conf. Ser.

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Abstract. To explore the role that temperature and percolation of rigidity play in determining the macroscopic properties, we propose a model that adds translational degrees of freedom to the spins of the well known Ising hamiltonian. In particular, the Ising model illustrate the longstanding idea that the growth of correlations on approach to a critical point could be describable in terms of the percolation of some sort of "physical cluster". For certain parameters of this model we observe two well defined peaks of CV , that suggest the existence of two kinds of "physical percolation", namely connectivity and rigidity percolation. Thermal fluctuations give rise to two different kinds of elementary excitations, i.e. droplets and configuron, as suggested by Angell in the framework of a bond lattice model approach. The later is reflected in the fluctuations of redundant constraints that gives stability to the structure and correlate with the order parameter. IntroductionIn order to explore the role that the lost of degrees of freedom (dof) plays due to the decrease of temperature in a magnetic network bond forming lattice, we propose an extension of the Ising Model. In this extension, each particle, that has a spin associated, and can freely translate inside a cell, but not escape from it. The later in a similar way to the early cell-theory of liquids, where each Wigner-Seitz cell is occupied by at most one particle [1], to mimic the conditions of a particle caged inside its neighbors [2]. Although mainly of academic interest, this model could have applications in the study of real systems as the spin-crossover in solid materials [3] or in the study of thermodynamical or structural properties of magnetic gels [4], where sudden sharp properties are mainly attributed to the percolation of the bond network [5,6]. For simplicity in this preliminary version of the model we have considered the case where all cells are occupied by one point-like particle (i.e. does not have excluded volume or area in 2D), then no interactions far from first neighbor cells can occurs, as we describe in the following.

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“…Models with lesser dilution [9,10] (number of bonds of order N ǫ with 1<ǫ<2 ) have been also considered, but in this case the thermodynamics is the same of the fully connected model.…”

Section: -The Dilute Ferromagnet and The Replica Solutionmentioning

confidence: 99%

Exact and approximate solutions for the dilute Ising model

Serva

2011

Physica A: Statistical Mechanics and its Applications

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The ground state energy and entropy of the dilute mean field Ising model is computed exactly by a single order parameter. An analogous exact solution is obtained in presence of a magnetic field with random locations. Results allow for a complete understanding of the geography of the associated random graph. In particular we give the size of the giant component (continent) and the number of isolated clusters of connected spins of all given size (islands). We also compute the average number of bonds per spin in the continent and in the islands. Then, we tackle the problem of computing the free energy of the dilute Ising model at strictly positive temperature. We are able to find out the exact solution in the paramagnetic region and exactly determine the phase transition line. In the ferromagnetic region we provide a solution in terms of an expansion with respect to a second parameter which can be made as accurate as necessary. All results are reached in the replica frame by a strategy which is not based on multi-overlaps.

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Finite size effects for the Ising model on random graphs with varying dilution

Cited by 7 publications

References 16 publications

Magnetization densities as replica parameters: The dilute ferromagnet

Magnetization densities as replica parameters: The dilute ferromagnet

Signature of Thermal Rigidity Percolation

Exact and approximate solutions for the dilute Ising model

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