Discrete approximations to vector spin models

We analyze a non-reversible mean-field jump dynamics for discrete qvalued rotators and show in particular that it exhibits synchronization. The dynamics is the mean-field analogue of the lattice dynamics investigated by the same authors in [26] which provides an example of a non-ergodic interacting particle system on the basis of a mechanism suggested by Maes and Shlosman [32].Based on the correspondence to an underlying model of continuous rotators via a discretization transformation we show the existence of a locally attractive periodic orbit of rotating measures. We also discuss global attractivity, using a free energy as a Lyapunov function and the linearization of the ODE which describes typical behavior of the empirical distribution vector.

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“…We call this solution ν ν ′ . The proof follows a line of arguments given in [14] in the lattice situation.…”

Section: Infinite-volume Rates: Existence and Non-existencementioning

confidence: 90%

Synchronization for discrete mean-field rotators

Jahnel

Külske

2014

Electron. J. Probab.

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“…That this is always possible follows from the earlier works [38,15]. More precisely, we assume that the condition from Theorem 2.1 of [15] is fulfilled (ensuring a regime where the Dobrushin uniqueness condition holds for the so-called constrained first-layer models where the Dobrushin condition is a weak dependence condition implying uniqueness and locality properties). Note, as in our notation the usual inverse temperature parameter β is incorporated into Φ, for β tending to infinity so does q 0 (Φ).…”

Section: The Equilibrium Modelmentioning

confidence: 96%

“…Step 1: We show that M τ,ν (η, σ Λ ) = M −τ (η, σ Λ ) for M τ ν-a.a. η (15) under the hypothesis of the theorem. To do so we extend the proof of the Gibbs variational principle in [27] Chapter 15.4.…”

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confidence: 92%

A class of non-ergodic probabilistic cellular automata with unique invariant measure and quasi-periodic orbit

Jahnel

Külske

2015

Stochastic Processes and their Applications

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We provide an example of a discrete-time Markov process on the threedimensional infinite integer lattice with Z q -invariant Bernoulli-increments which has as local state space the cyclic group Z q . We show that the system has a unique invariant measure, but remarkably possesses an invariant set of measures on which the dynamics is conjugate to an irrational rotation on the continuous sphere S 1 . The update mechanism we construct is exponentially well localized on the lattice.

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“…(For more results around the issue of differences and similarities between continuous spin-models and clock models see e.g. [12,13,19,27].) This paper consists of two main parts:…”

Section: Introductionmentioning

confidence: 99%

Non-robust Phase Transitions in the Generalized Clock Model on Trees

Külske¹,

Philipp²

2017

J Stat Phys

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Abstract. Pemantle and Steif provided a sharp threshold for the existence of a RPT (robust phase transition) for the continuous rotator model and the Potts model in terms of the branching number and the second eigenvalue of the transfer operator, where a robust phase transition is said to occur if an arbitrarily weak coupling with symmetrybreaking boundary conditions suffices to induce symmetry breaking in the bulk. They further showed that for the Potts model RPT occurs at a different threshold than PT (phase transition in the sense of multiple Gibbs measures), and conjectured that RPT and PT should occur at the same threshold in the continuous rotator model.We consider the class of 4-and 5-state rotation-invariant spin models with reflection symmetry on general trees which contains the Potts model and the clock model with scalarproduct-interaction as limiting cases. The clock model can be viewed as a particular discretization which is obtained from the classical rotator model with state space S 1 . We analyze the transition between PT=RPT and PT =RPT, in terms of the eigenvalues of the transfer matrix of the model at the critical threshold value for the existence of RPT. The transition between the two regimes depends sensitively on the third largest eigenvalue.Mathematics Subject Classifications (2010). 82B26 (primary); 60K35 (secondary)

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Discrete approximations to vector spin models

Cited by 14 publications

References 28 publications

Synchronization for discrete mean-field rotators

Synchronization for discrete mean-field rotators

A class of non-ergodic probabilistic cellular automata with unique invariant measure and quasi-periodic orbit

Non-robust Phase Transitions in the Generalized Clock Model on Trees

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