Sharp thresholds for Gibbs-non-Gibbs transition in the fuzzy Potts model with a Kac-type interaction

We investigate the Gibbs properties of the fuzzy Potts model on the d-dimensional torus with Kac interaction. We use a variational approach for profiles inspired by that of Fernandez, den Hollander and Mart{i}nez for their study of the Gibbs-non-Gibbs transitions of a dynamical Kac-Ising model on the torus. As our main result, we show that the mean-field thresholds dividing Gibbsian from non-Gibbsian behavior are sharp in the fuzzy Kac-Potts model with class size unequal two. On the way to this result we prove a large deviation principle for color profiles with diluted total mass densities and use monotocity arguments.

Synchronization for discrete mean-field rotators

We analyze a non-reversible mean-field jump dynamics for discrete q-valued 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.

Discrete approximations to vector spin models

We strengthen a result of two of us on the existence of effective interactions for discretised continuous-spin models. We also point out that such an interaction cannot exist at very low temperatures. Moreover, we compare two ways of discretising continuous-spin models, and show that, except for very low temperatures, they behave similarly in two dimensions. We also discuss some possibilities in higher dimensions.

A symmetric entropy bound on the non-reconstruction regime of Markov chains on Galton-Watson trees

We give a criterion of the form Q(d)c(M)<1 for the non-reconstructability of tree-indexed q-state Markov chains obtained by broadcasting a signal from the root with a given transition matrix M. Here c(M) is an explicit function, which is convex over the set of Ms with a given invariant distribution, that is defined in terms of a (q-1)-dimensional variational problem over symmetric entropies. Further Q(d) is the expected number of offspring on the Galton-Watson tree. This result is equivalent to proving the extremality of the free boundary condition-Gibbs measure within the corresponding Gibbs-simplex. Our theorem holds for possibly non-reversible M and its proof is based on a general Recursion Formula for expectations of a symmetrized relative entropy function, which invites their use as a Lyapunov function. In the case of the Potts model, the present theorem reproduces earlier results of the authors, with a simplified proof, in the case of the symmetric Ising model (where the argument becomes similar to the approach of Pemantle and Peres) the method produces the correct reconstruction threshold), in the case of the (strongly) asymmetric Ising model where the Kesten-Stigum bound is known to be not sharp the method provides improved numerical bounds.

On the Purity of the free boundary condition Potts measure on random trees

We consider the free boundary condition Gibbs measure of the Potts model on a random tree. We provide an explicit temperature interval below the ferromagnetic transition temperature for which this measure is extremal, improving older bounds of Mossel and Peres. In information theoretic language extremality of the Gibbs measure corresponds to non-reconstructability for symmetric q-ary channels. The bounds are optimal for the Ising model and appear to be close to what we conjecture to be the true values up to a factor of 0.0150 in the case q = 3 and 0.0365 for q = 4. Our proof uses an iteration of random boundary entropies from the outside of the tree to the inside, along with a symmetrization argument.

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.

A second row Parking Paradox

We consider two variations of the discrete car parking problem where at every vertex of the integers a car arrives with rate one, now allowing for parking in two lines. a) The car parks in the first line whenever the vertex and all of its nearest neighbors are not occupied yet. It can reach the first line if it is not obstructed by cars already parked in the second line (screening). b) The car parks according to the same rules, but parking in the first line can not be obstructed by parked cars in the second line (no screening). In both models, a car that can not park in the first line will attempt to park in the second line. If it is obstructed in the second line as well, the attempt is discarded. We show that both models are solvable in terms of finite-dimensional ODEs. We compare numerically the limits of first and second line densities, with time going to infinity. While it is not surprising that model a) exhibits an increase of the density in the second line from the first line, more remarkably this is also true for model b), albeit in a less pronounced way.

Continuous Spin Mean-Field models: Limiting kernels and Gibbs Properties of local transforms

We extend the notion of Gibbsianness for mean-field systems to the set-up of general (possibly continuous) local state spaces. We investigate the Gibbs properties of systems arising from an initial mean-field Gibbs measure by application of given local transition kernels. This generalizes previous case-studies made for spins taking finitely many values to the first step in direction to a general theory, containing the following parts: (1) A formula for the limiting conditional probability distributions of the transformed system. It holds both in the Gibbs and non-Gibbs regime and invokes a minimization problem for a constrained rate-function. (2) A criterion for Gibbsianness of the transformed system for initial Lipschitz-Hamiltonians involving concentration properties of the transition kernels. (3) A continuity estimate for the single-site conditional distributions of the transformed system. While (2) and (3) have provable lattice-counterparts, the characterization of (1) is stronger in mean-field. As applications we show short-time Gibbsianness of rotator mean-field models on the (q-1)-dimensional sphere under diffusive time-evolution and the preservation of Gibbsianness under local coarse-graining of the initial local spin space.

Parking on a Random Tree

Consider an infinite tree with random degrees, i.i.d. over the sites, with a prescribed probability distribution with generating function G(s). We consider the following variation of Renyis parking problem, alternatively called blocking RSA: at every vertex of the tree a particle (or car) arrives with rate one. The particle sticks to the vertex whenever the vertex and all of its nearest neighbors are not occupied yet. We provide an explicit expression for the so-called parking constant in terms of the generating function.

The Posterior metric and the Goodness of Gibbsianness for transforms of Gibbs measures

We present a general method to derive continuity estimates for conditional probabilities of general (possibly continuous) spin models sub jected to local transformations. Such systems arise in the study of a stochastic time-evolution of Gibbs measures or as noisy observations. We exhibit the minimal necessary structure for such double-layer systems. Assuming no a priori metric on the local state spaces, we define the posterior metric on the local image space. We show that it allows in a natural way to divide the local part of the continuity estimates from the spatial part (which is treated by Dobrushin uniqueness here). We show in the concrete example of the time evolution of rotators on the q-1 dimensional sphere how this method can be used to obtain estimates in terms of the familiar Euclidean metric.