Wioletta M. Ruszel

Gibbs-non-Gibbs properties for n-vector lattice and mean-field models

We review some recent developments in the study of Gibbs and non-Gibbs properties of transformed n-vector lattice and mean-field models under various transformations. Also, some new results for the loss and recovery of the Gibbs property of planar rotor models during stochastic time evolution are presented.

Duality and stationary distributions of wealth distribution models

We analyze a class of energy and wealth redistribution models, characterizing their stationary measures and showing that they have a discrete dual process. In particular we show that the wealth distribution model with non-zero saving propensity can never have invariant product measures. We also introduce diffusion processes associated to the wealth distribution models by ‘instantaneous thermalization’.

Effect of self-interaction on the phase diagram of a Gibbs-like measure derived by a reversible Probabilistic Cellular Automata

Abstract Cellular Automata are discrete-time dynamical systems on a spatially extended discrete space which provide paradigmatic examples of nonlinear phenomena. Their stochastic generalizations, i.e., Probabilistic Cellular Automata (PCA), are discrete time Markov chains on lattice with finite single-cell states whose distinguishing feature is the parallel character of the updating rule. We study the ground states of the Hamiltonian and the low-temperature phase diagram of the related Gibbs measure naturally associated with a class of reversible PCA, called the cross PCA. In such a model the updating rule of a cell depends indeed only on the status of the five cells forming a cross centered at the original cell itself. In particular, it depends on the value of the center spin (self-interaction). The goal of the paper is that of investigating the role played by the self-interaction parameter in connection with the ground states of the Hamiltonian and the low-temperature phase diagram of the Gibbs measure associated with this particular PCA.

Scaling limit of the odometer in divisible sandpiles

In a recent work Levine et al. (Ann Henri Poincaré 17:1677–1711, 2016. https://doi.org/10.1007/s00023-015-0433-x) prove that the odometer function of a divisible sandpile model on a finite graph can be expressed as a shifted discrete bilaplacian Gaussian field. For the discrete torus, they suggest the possibility that the scaling limit of the odometer may be related to the continuum bilaplacian field. In this work we show that in any dimension the rescaled odometer converges to the continuum bilaplacian field on the unit torus.

Strongly reinforced pólya urns with graph-based competition

We introduce a class of reinforcement models where, at each time step

Loss and recovery of Gibbsianness for XY models in external fields

t

Contour Methods for Long-Range Ising Models: Weakening Nearest-Neighbor Interactions and Adding Decaying Fields

, one first chooses a random subset

Non-criticality criteria for Abelian sandpile models with sources and sinks

A_t

Synchronization and Spin-Flop Transitions for a Mean-Field XY Model in Random Field

of colours (independent of the past) from

Multilinearity of the covariances in one-dimensional models out of equilibrium

n