zbMATH — the first resource for mathematics

A survey of dynamical percolation. (English) Zbl 1186.60106
Bandt, Christoph (ed.) et al., Fractal geometry and stochastics IV. Proceedings of the 4th conference, Greifswald, Germany, September 8–12, 2008. Basel: Birkhäuser (ISBN 978-3-0346-0029-3/hbk; 978-3-0346-0030-9/ebook). Progress in Probability 61, 145-174 (2009).
The paper under review is a well-written survey article about the theory of dynamical percolation. Without going too much into details it provides a good guide to the relevant literature (54 items). The concept of dynamical percolation was introduced by O. Häggström, Y. Peres and J. E. Steif [Ann. Inst. H. Poincaré Probab. Statist. 33, No. 4, 497–528 (1997; Zbl 0894.60098)]. Dynamical percolation is a continuous time (two states) Markov chain on the space of subgraphs of a given infinite connected locally finite graph that has the usual percolation measure as its stationary distribution. One is interested in properties of the random walk that are dynamically stable, i.e. hold for all times a.s., and in dynamically sensitive properties being not valid on a set of exceptional times. The general question in dynamical percolation is to investigate dynamical stability resp. sensitivity of percolation properties. The (slightly shortened) table of contents provides a good view of the paper:
1. Introduction, 2. Dynamical percolation: first results, 3. Exceptional times of percolation: tree case, 4. Noise sensitivity, noise stability, 5. Critical exponents for percolation, 6. Exceptional times for the hexagonal lattice, 7. The exact Hausdorff dimension of exceptional times, 8. Sensitivity of the infinite cluster in critical percolation, 9. Dynamical percolation and the incipient infinite cluster, 10. The scaling limit of planar dynamical percolation, 11. Dynamical percolation for interacting particle systems.
For the entire collection see [Zbl 1173.37002].

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C43 Time-dependent percolation in statistical mechanics
82C27 Dynamic critical phenomena in statistical mechanics
82B43 Percolation
82B26 Phase transitions (general) in equilibrium statistical mechanics
Full Text: arXiv