Graf, Robert (2014): Forestfire models and their critical limits. Dissertation, LMU München: Faculty of Mathematics, Computer Science and Statistics 

PDF
Graf_Robert.pdf 16MB 
Abstract
Forestfire processes were first introduced in the physics literature as a toy model for selforganized criticality. The term selforganized criticality describes interacting particle systems which are governed by local interactions and are inherently driven towards a perpetual critical state. As in equilibrium statistical physics, the critical state is characterized by longrange correlations, power laws, fractal structures and selfsimilarity. We study several different forestfire models, whose common features are the following: All models are continuoustime processes on the vertices of some graph. Every vertex can be "vacant" or "occupied by a tree". We start with some initial configuration. Then the process is governed by two competing random mechanisms: On the one hand, vertices become occupied according to rate 1 Poisson processes, independently of one another. On the other hand, occupied clusters are "set on fire" according to some predefined rule. In this case the entire cluster is instantaneously destroyed, i.e. all of its vertices become vacant. The selforganized critical behaviour of forestfire models can only occur on infinite graphs such as planar lattices or infinite trees. However, in all relevant versions of forestfire models, the destruction mechanism is a priori only welldefined for finite graphs. For this reason, one starts with a forestfire model on finite subsets of an infinite graph and then takes the limit along increasing sequences of finite subsets to obtain a new forestfire model on the infinite graph. In this thesis, we perform this kind of limit for two classes of forestfire models and investigate the resulting limit processes.
Item Type:  Theses (Dissertation, LMU Munich) 

Keywords:  forestfire model, selforganized criticality 
Subjects:  500 Natural sciences and mathematics 500 Natural sciences and mathematics > 510 Mathematics 
Faculties:  Faculty of Mathematics, Computer Science and Statistics 
Language:  English 
Date of oral examination:  15. December 2014 
1. Referee:  Merkl, Franz 
MD5 Checksum of the PDFfile:  1fbebc8761c64aff9f52e08d480f26b5 
Signature of the printed copy:  0001/UMC 22619 
ID Code:  17780 
Deposited On:  20. Jan 2015 13:46 
Last Modified:  23. Oct 2020 22:37 