Asymptotic analysis of a ferromagnetic Ising system with “diffuse” interfacial energy

Braides, Andrea; Causin, Andrea; Solci, Margherita

doi:10.1007/s10231-017-0693-9

Cited by 3 publications

(2 citation statements)

References 16 publications

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“…For sparse graphs (i.e., graphs which are not dense according to the definition above) and interactions not satisfying the decay conditions ( 6), the correct scaling, the relative convergence and the form of the Γ-limit is a complex open problem. In [8] an example is given of one-dimensional energies with range R ε = 1/ √ ε such that a non-trivial Γ-limit exists for s ε = 1/ √ ε with respect to the L ∞ -weak * convergence, but it is defined on all functions of bounded variation with values in [0, 1] (and not only those with values in {0, 1}). In that example a crucial issue is the topology of the graph of the connections where a ε ij = 0.…”

Section: Introductionmentioning

confidence: 99%

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Compactness by coarse-graining in long-range lattice systems

Braides¹,

Solci²

2019

Preprint

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We consider energies on a periodic set L of the form i,j∈L a ε ij |u i − u j |, defined on spin functions u i ∈ {0, 1}, and we suppose that the typical range of the interactions isIn a discrete-tocontinuum analysis, we prove that the overall behaviour as ε → 0 of such functionals is that of an interfacial energy. The proof is performed using a coarse-graining procedure which associates to scaled functions defined on εL with equibounded energy a family of sets with equibounded perimeter. This agrees with the case of equibounded R ε and can be seen as an extension of coerciveness result for short-range interactions, but is different from that of other long-range interaction energies, whose limit exits the class of surface energies. A computation of the limit energy is performed in the case L = Z d .

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Section: Introductionmentioning

confidence: 99%

“…Margherita Solci acknowledges the project "Fondo di Ateneo per la ricerca 2019", funded by the University of Sassari. We thank the anonymous referee of [8], who drew our attention to the problem in this paper.…”

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Compactness by coarse-graining in long-range lattice systems

Braides¹,

Solci²

2019

Preprint

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Γ-limit of the cut functional on dense graph sequences

2020

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A sequence of graphs with diverging number of nodes is a dense graph sequence if the number of edges grows approximately as for complete graphs. To each such sequence a function, called graphon, can be associated, which contains information about the asymptotic behavior of the sequence. Here we show that the problem of subdividing a large graph in communities with a minimal amount of cuts can be approached in terms of graphons and the Γ-limit of the cut functional, and discuss the resulting variational principles on some examples. Since the limit cut functional is naturally defined on Young measures, in many instances the partition problem can be expressed in terms of the probability that a node belongs to one of the communities. Our approach can be used to obtain insights into the bisection problem for large graphs, which is known to be NP-complete.

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Compactness by Coarse-Graining in Long-Range Lattice Systems

Braides

Solci

2020

Advanced Nonlinear Studies

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We consider energies on a periodic set ℒ of the form \sum_{i,j\in\mathcal{L}}a^{\varepsilon}_{ij}\lvert u_{i}-u_{j}\rvert, defined on spin functions u_{i}\in\{0,1\}, and we suppose that the typical range of the interactions is R_{\varepsilon} with R_{\varepsilon}\to+\infty, i.e., if \lvert i-j\rvert\leq R_{\varepsilon}, then a^{\varepsilon}_{ij}\geq c>0. In a discrete-to-continuum analysis, we prove that the overall behavior as \varepsilon\to 0 of such functionals is that of an interfacial energy. The proof is performed using a coarse-graining procedure which associates to scaled functions defined on \varepsilon\mathcal{L} with equibounded energy a family of sets with equibounded perimeter. This agrees with the case of equibounded R_{\varepsilon} and can be seen as an extension of coerciveness result for short-range interactions, but is different from that of other long-range interaction energies, whose limit exits the class of surface energies. A computation of the limit energy is performed in the case \mathcal{L}=\mathbb{Z}^{d}.

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Asymptotic analysis of a ferromagnetic Ising system with “diffuse” interfacial energy

Cited by 3 publications

References 16 publications

Compactness by coarse-graining in long-range lattice systems

Compactness by coarse-graining in long-range lattice systems

Γ-limit of the cut functional on dense graph sequences

Compactness by Coarse-Graining in Long-Range Lattice Systems

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