Projects / Programmes
Cellular automata and percolation models.
| Code |
Science |
Field |
Subfield |
| 1.01.01 |
Natural sciences and mathematics |
Mathematics |
Analysis |
| Code |
Science |
Field |
| P001 |
Natural sciences and mathematics |
Mathematics |
cellular automaton, percolation, nucleation, asymptotic shape, hydrodynamic limit.
Researchers (2)
| no. |
Code |
Name and surname |
Research area |
Role |
Period |
No. of publicationsNo. of publications |
| 1. |
04997 |
PhD Janko Gravner |
Mathematics |
Head |
1999 - 2000 |
0 |
| 2. |
10013 |
PhD Mihael Perman |
Mathematics |
Researcher |
1998 - 2000 |
0 |
Organisations (1)
Abstract
This research grant is devoted to studying several deterministic and random cellular automata models as well as percolation problems and random graphs. Inspiration for such models comes form processes in nature and emphasis is on developing mathematical techniques for their analysis. The main undelying theme is to understand the connection between a simple local rule and the resulting global phenomena. The local rules are usually suggested by physical principles, and they attempt to capture at least some essential aspect of the underlying process. Although higly idealized, such models often exibits behaviour which is strikingly similar to that observed in nature, thus helping to understand how simple mechanisms can give rise to complex patterns. The main themes of this research are described below: (1) In many natural processes, one equilibrium displaces another by nucleation, that is, by finding rare local configurations, which are able to to initiate growth which then proceeds without bounds. We study simple cellular automata (CA) models which exibit such behaviour: in particular, we address density of critical configurations, shapes and interactions of growing droplets, and response to pollution in space. (2) Evolution by surface tension is an important phenomenon in physics. In many other contexts, competition between two equally strong eqilibria (e.g. equally fit species) results in such dynamics as well. The majority vote CA is perhaps the simplest model which exibits such behaviour. For the infinite system, recently established PDE tools offer good prospects to prove clustering theorems in the case of random symmetric initial states with smooth boundaries. For finite systems, the properties of final state depend on topological and geometric properties of the system. (3) Investigations described above are by nature low dimensional, limited to two or at most three space dimensions. On the other hand, biological genotypes form a high dimensional space to which a high-dimensional binary hypercube is a good approximation and in which the nearest neighbour connections can be interpreted as mutations. We study connectivity properties of random sets of genotypes, trying to understand the mechanisms which relate diversity of species with genetic constraints and environmental conditions. (4) Finally, part of this investigation is experimental, developing computer tools for simulations, combinatorial enumarations and optimizations, and numerical computations arising from mathematical analysis of CA models.
