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Stationary Measures and Phase Transition for a Class of Probabilistic Cellular Automata
"... We discuss various properties of Probabilistic Cellular Automata, such as the structure of the set of stationary measures and multiplicity of stationary measures (or phase transition) for reversible models. ..."
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We discuss various properties of Probabilistic Cellular Automata, such as the structure of the set of stationary measures and multiplicity of stationary measures (or phase transition) for reversible models.
The Restriction of the Ising Model to a Layer
, 1998
"... We discuss the status of recent Gibbsian descriptions of the restriction (projection) of the Ising phases to a layer. We concentrate on the projection of the twodimensional low temperature Ising phases for which we prove a variational principle. ..."
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We discuss the status of recent Gibbsian descriptions of the restriction (projection) of the Ising phases to a layer. We concentrate on the projection of the twodimensional low temperature Ising phases for which we prove a variational principle.
Cardy’s formula for some dependent percolation models
 Bull. Brazilian Math. Soc
, 2002
"... We prove Cardy’s formula for rectangular crossing probabilities in dependent site percolation models that arise from a deterministic cellular automaton with a random initial state. The cellular automaton corresponds to the zerotemperature case of Domany’s stochastic Ising ferromagnet on the hexagon ..."
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We prove Cardy’s formula for rectangular crossing probabilities in dependent site percolation models that arise from a deterministic cellular automaton with a random initial state. The cellular automaton corresponds to the zerotemperature case of Domany’s stochastic Ising ferromagnet on the hexagonal lattice H (with alternating updates of two sublattices) [7]; it may also be realized on the triangular lattice T with flips when a site disagrees with six, five and sometimes four of its six neighbors. 1
A meaningful expansion around detailed balance
 J. Phys. A
"... Abstract. We consider Markovian dynamics modeling open mesoscopic systems which are driven away from detailed balance by a nonconservative force. A systematic expansion is obtained of the stationary distribution around an equilibrium reference, in orders of the nonequilibrium forcing. The first orde ..."
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Abstract. We consider Markovian dynamics modeling open mesoscopic systems which are driven away from detailed balance by a nonconservative force. A systematic expansion is obtained of the stationary distribution around an equilibrium reference, in orders of the nonequilibrium forcing. The first order around equilibrium has been known since the work of McLennan (1959), and involves the transient irreversible entropy flux. The expansion generalizes the McLennan formula to higher orders, complementing the entropy flux with the dynamical activity. The latter is more kinetic than thermodynamic and is a possible realization of Landauer’s insight (1975) that, for nonequilibrium, the relative occupation of states also depends on the noise along possible escape routes. In that way nonlinear response around equilibrium can be meaningfully discussed in terms of two main quantities only, the entropy flux and the dynamical activity. The expansion makes mathematical sense as shown in the simplest cases from exponential ergodicity.
A Particular Bit of Universality: Scaling Limits of Some Dependent Percolation Models
 Comm. Math. Phys
, 2004
"... We study families of dependent site percolation models on the triangular lattice T and hexagonal lattice H that arise by applying certain cellular automata to independent percolation configurations. We analyze the scaling limit of such models and show that the distance between macroscopic portions o ..."
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We study families of dependent site percolation models on the triangular lattice T and hexagonal lattice H that arise by applying certain cellular automata to independent percolation configurations. We analyze the scaling limit of such models and show that the distance between macroscopic portions of cluster boundaries of any two percolation models within one of our families goes to zero almost surely in the scaling limit. It follows that each of these cellular automaton generated dependent percolation models has the same scaling limit (in the sense of AizenmanBurchard [3]) as independent site percolation on T.
Dualities for MultiState Probabilistic Cellular Automata
, 2006
"... The present work treats dualities for probabilistic cellular automata (PCA). A general result of duality is presented and it is used to study two models of PCA: a multiopinion noisy general voter model; and a multistate attractive biased model. 1 ..."
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The present work treats dualities for probabilistic cellular automata (PCA). A general result of duality is presented and it is used to study two models of PCA: a multiopinion noisy general voter model; and a multistate attractive biased model. 1
A selection of nonequilibrium issues
 In Lecture notes in Mathematics
"... Summary. We give a pedagogical introduction to a selection of recently discussed topics in nonequilibrium statistical mechanics, concentrating mostly on formal structures and on general principles. Part I contains an overview of the formalism of lattice gases that we use to explain various symmetrie ..."
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Summary. We give a pedagogical introduction to a selection of recently discussed topics in nonequilibrium statistical mechanics, concentrating mostly on formal structures and on general principles. Part I contains an overview of the formalism of lattice gases that we use to explain various symmetries and inequalities generally valid for nonequilibrium systems, including the fluctuation symmetry, Jarzynski equality, and the direction of currents. That mostly concerns the timeantisymmetric part of dynamical fluctuation theory. We also briefly comment on recent attempts to combine that with the timesymmetric sector in a Langrangian or extended OnsagerMachlup approach. In Part II we concentrate on the macroscopic state and how entropy provides a bridge between microscopic dynamics and macroscopic irreversibility; included is a construction of quantum macroscopic states and a result on the equivalence of ensembles. Part I. Fluctuations in stochastic lattice gases 1
Critical droplets in metastable states of probabilistic cellular automata
 PHYS. REV. E
, 1999
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Percolation, Path Large Deviations and Weakly Gibbs States
, 1998
"... We present a unified approach to establishing the Gibbsian character of a wide class of nonGibbsian states, arising in the Renormalisation Group theory. Inside the realm of the PirogovSinai theory for lattice spin systems, we prove that RG transformations applied to low temperature phases give ris ..."
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We present a unified approach to establishing the Gibbsian character of a wide class of nonGibbsian states, arising in the Renormalisation Group theory. Inside the realm of the PirogovSinai theory for lattice spin systems, we prove that RG transformations applied to low temperature phases give rise to weakly Gibbsian measures. In other words, we show that the GriffithsPearceIsrael scenario of RG pathologies is carried by atypical configurations. The renormalized measures are described by an effective interaction, with relative energies welldefined on a full measure set of configurations. In this way we complete the first part of the Dobrushin Restoration Program: to give a Gibbsian description to nonGibbsian states. A disagreement percolation estimate is used in the proof to bound the decay of