The Renormalization-Group peculiarities of Griffiths and Pearce: What have we learned?

We review what we have learned about the "Renormalization-Group peculiarities" which were discovered about twenty years ago by Griffiths and Pearce, and which questions they asked are still widely open. We also mention some related developments.Comment: Proceedings Marseille meeting on mathematical results in statistical mechanic

Scaling and Inverse Scaling in Anisotropic Bootstrap percolation

In bootstrap percolation it is known that the critical percolation threshold tends to converge slowly to zero with increasing system size, or, inversely, the critical size diverges fast when the percolation probability goes to zero. To obtain higher-order terms (that is, sharp and sharper thresholds) for the percolation threshold in general is a hard question. In the case of two-dimensional anisotropic models, sometimes correction terms can be obtained from inversion in a relatively simple manner.Comment: Contribution to the proceedings of the 2013 EURANDOM workshop Probabilistic Cellular Automata: Theory, Applications and Future Perspectives, equation typo corrected, constant of generalisation correcte

Aperiodicity in equilibrium systems: Between order and disorder

Spatial aperiodicity occurs in various models and material s. Although today the most well-known examples occur in the area of quasicrystals, other applications might also be of interest. Here we discuss some issues related to the notion and occurrence of aperiodic order in equilibrium statistical mechanics. In particular, we consider some spectral characterisations,and shortly review what is known about the occurrence of aperiodic order in lattice models at zero and non-zero temperatures. At the end some more speculative connections to the theory of (spin-)glasses are indicated.Comment: Contribution to ICQ12, some corrections and explanatory remarks adde

A remark on the notion of robust phase transitions

We point out that the high-q Potts model on a regular lattice at its transition temperature provides an example of a non-robust - in the sense recently proposed by Pemantle and Steif- phase transition

Nonexistence of random gradient Gibbs measures in continuous interface models in d=2d=2

We consider statistical mechanics models of continuous spins in a disordered environment. These models have a natural interpretation as effective interface models. It is well known that without disorder there are no interface Gibbs measures in infinite volume in dimension $d=2$, while there are ``gradient Gibbs measures'' describing an infinite-volume distribution for the increments of the field, as was shown by Funaki and Spohn. In the present paper we show that adding a disorder term prohibits the existence of such gradient Gibbs measures for general interaction potentials in $d=2$. This nonexistence result generalizes the simple case of Gaussian fields where it follows from an explicit computation. In $d=3$ where random gradient Gibbs measures are expected to exist, our method provides a lower bound of the order of the inverse of the distance on the decay of correlations of Gibbs expectations w.r.t. the distribution of the random environment.Comment: Published in at http://dx.doi.org/10.1214/07-AAP446 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Multiple-Layer Parking with Screening

In this article a multilayer parking system with screening of size n=3 is studied with a focus on the time-dependent particle density. We prove that the asymptotic limit of the particle density increases from an average density of 1/3 on the first layer to the value of (10 - \sqrt 5 )/19 in higher layers

Scaling and Inverse Scaling in Anisotropic Bootstrap Percolation

In bootstrap percolation, it is known that the critical percolation threshold tends to converge slowly to zero with increasing system size, or, inversely, the critical size diverges fast when the percolation probability goes to zero. To obtain higher-order terms (i.e. sharp and sharper thresholds) for the percolation threshold in general is a hard question. In the case of two-dimensional anisotropic models, sometimes such correction terms can be obtained from inversion in a relatively simple manner