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.
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 occurren
ce 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.
in this article a multilayer parking system of size n=3 is studied. We prove that the asymptotic limit of the particle density in the center approaches a maximum of 1/2 in higher layers. This means a significant increase of capacity compared to the f
irst layer where this value is 1/3. This is remarkable because the process is solely driven by randomness. We conjecture that the results applies to all finite parking systems with n larger or equal than 2.
