Criticality of Measures on 2-d Ising Configurations: From Square to Hexagonal Graphs

Apollonio, Valentina; D'Autilia, Roberto; Scoppola, Benedetto; Scoppola, Elisabetta; Troiani, Alessio

doi:10.1007/s10955-019-02403-3

Cited by 9 publications

(20 citation statements)

References 11 publications

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“…The parameter q is also referred to as the self interaction parameter. As pointed out in [1] a careful look to the Hamiltonian (5) and to the graph of Fig. 1 shows that the bipartite graph is isomorphic to the hexagonal lattice G (V , E) with edges J and q on whose vertices are arranged the variables σ and τ as shown in Fig.…”

Section: Consider the Ising Hamiltonian On A Graphmentioning

confidence: 94%

“…. , σ k ), where τ i , σ i ∈ {−1, 1} for each i, and are arranged on a bipartite graph [1]. Different interactions among the τ and σ variables give rise to the possibility of interpolation among different lattice geometries.…”

Section: Introductionmentioning

confidence: 99%

“…Different interactions among the τ and σ variables give rise to the possibility of interpolation among different lattice geometries. The equilibrium measure of the shaken dynamics has been extensively investigated and the analytic form of the critical curve in the (q, J ) parameters space has been found [1].…”

Section: Introductionmentioning

confidence: 99%

“…The PCA dynamics we consider is a parallel and irreversible version of the heath bath dynamics and is obtained by concatenating two different update rules [2] (shaken dynamics). By the Hamiltonian defined in [1], which depends on the (J , q) parameters, we identify numerically two regions of the space (J , q) characterized by different behaviors of the dynamics.…”

Section: Introductionmentioning

confidence: 99%

“…The elementary step of the shaken dynamics is naturally defined on the a finite subset of the square lattice Z 2 and consists of a sequence of two inhomogeneous half steps. However, in both [1,2] it has been pointed out that the shaken dynamics can be seen as an alternate dynamics on a subset of the honeycomb lattice. The proposed dynamics, although not faster than ad hoc dynamics for specific models, simulates a whole class of statistical mechanics models spanning from the one-dimensional Ising model to the square lattice and hexagonal one across all the intermediate models depending on the values of J and q.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Parallel Simulation of Two-Dimensional Ising Models Using Probabilistic Cellular Automata

2021

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We study the numerical simulation of the shaken dynamics, a parallel Markovian dynamics for spin systems with local interaction and transition probabilities depending on the two parameters q and J that “tune” the geometry of the underlying lattice. The analysis of the mixing time of the Markov chain and the evaluation of the spin-spin correlations as functions of q and J, make it possible to determine in the (q, J) plane a phase transition curve separating the disordered phase from the ordered one. The relation between the equilibrium measure of the shaken dynamics and the Gibbs measure for the Ising model is also investigated. Finally two different coding approaches are considered for the implementation of the dynamics: a multicore CPU approach, coded in Julia, and a GPU approach coded with CUDA.

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Section: Consider the Ising Hamiltonian On A Graphmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Parallel Simulation of Two-Dimensional Ising Models Using Probabilistic Cellular Automata

2021

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Shaken Dynamics: An Easy Way to Parallel Markov Chain Monte Carlo

et al. 2022

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We define a class of Markovian parallel dynamics for spin systems on arbitrary graphs with nearest neighbor interaction described by a Hamiltonian function $$H(\sigma )$$ H ( σ ) . These dynamics turn out to be reversible and their stationary measure is explicitly determined. Convergence to equilibrium and relation of the stationary measure to the usual Gibbs measure are discussed when the dynamics is defined on $$\mathbb {Z}^2$$ Z 2 . Further it is shown how these dynamics can be used to define natively parallel algorithms to face problems in the context of combinatorial optimization.

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Metastability for Kawasaki Dynamics on the Hexagonal Lattice

Baldassarri

Jacquier

2023

J Stat Phys

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In this paper we analyze the metastable behavior for the Ising model that evolves under Kawasaki dynamics on the hexagonal lattice $${\mathbb {H}}^2$$ H 2 in the limit of vanishing temperature. Let $$\varLambda \subset {\mathbb {H}}^2$$ Λ ⊂ H 2 a finite set which we assume to be arbitrarily large. Particles perform simple exclusion on $$\varLambda $$ Λ , but when they occupy neighboring sites they feel a binding energy $$-U<0$$ - U < 0 . Along each bond touching the boundary of $$\varLambda $$ Λ from the outside to the inside, particles are created with rate $$\rho =e^{-\varDelta \beta }$$ ρ = e - Δ β , while along each bond from the inside to the outside, particles are annihilated with rate 1, where $$\beta $$ β is the inverse temperature and $$\varDelta >0$$ Δ > 0 is an activity parameter. For the choice $$\varDelta \in {(U,\frac{3}{2}U)}$$ Δ ∈ ( U , 3 2 U ) we prove that the empty (resp. full) hexagon is the unique metastable (resp. stable) state. We determine the asymptotic properties of the transition time from the metastable to the stable state and we give a description of the critical configurations. We show how not only their size but also their shape varies depending on the thermodynamical parameters. Moreover, we emphasize the role that the specific lattice plays in the analysis of the metastable Kawasaki dynamics by comparing the different behavior of this system with the corresponding system on the square lattice.

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Criticality of Measures on 2-d Ising Configurations: From Square to Hexagonal Graphs

Cited by 9 publications

References 11 publications

Parallel Simulation of Two-Dimensional Ising Models Using Probabilistic Cellular Automata

Parallel Simulation of Two-Dimensional Ising Models Using Probabilistic Cellular Automata

Shaken Dynamics: An Easy Way to Parallel Markov Chain Monte Carlo

Metastability for Kawasaki Dynamics on the Hexagonal Lattice

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