Matzke, Kilian (2020): On phase transitions in random spatial systems. Dissertation, LMU München: Faculty of Mathematics, Computer Science and Statistics 

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Abstract
This dissertation is concerned with the analysis of three models, all of which have their motivation in the field of statistical mechanics. The first model (investigated in the first contribution) is an interacting particle system with three states, while the other two models (investigated in the second, third, and fourth contribution) are percolation models. For the first model, we confirm the existence of a phase transition. In the second and third model, we investigate behavior of the models at (and close to) the point of the phase transition. In the first contribution, the particles are indexed by Z and can be in one of three states, one of which we call "infected". Interactions take place with the two nearestneighbor particles and according to some rule governed by a parameter q [0,1]. We investigate survival (for large times) of the set of infected particles and prove that for q small enough, the infection survives with positive probability, whereas for q close enough to 1, it almost surely dies outhence, between the different regimes of $q$, a phase transition occurs. In the second and third contribution, the wellknown site percolation model is considered in high dimensions. We derive the lace expansion, an identity for the model's twopoint function, to deduce the triangle condition, the infrared bound, and thus meanfield behavior in sufficiently high dimension. The third contribution then builds on this derived lace expansion to explicitly compute the first terms of the asymptotic expansion of the critical point pc. The fourth contribution investigates the random connection model, which can be viewed as a generalization of the continuum analogue of site percolation. We investigate the model's meanfield behavior in high dimension through a continuumspace adaption of the lace expansion. Moreover, as an example of the implied meanfield behavior, the critical exponent $\gamma$ is proven to exist and (as on the lattice) to take value 1.
Item Type:  Theses (Dissertation, LMU Munich) 

Subjects:  300 Social sciences 300 Social sciences > 310 General statistics 
Faculties:  Faculty of Mathematics, Computer Science and Statistics 
Language:  English 
Date of oral examination:  30. June 2020 
1. Referee:  Heydenreich, Markus 
MD5 Checksum of the PDFfile:  9f94f48140fb9909e097555299b7613f 
Signature of the printed copy:  0001/UMC 27407 
ID Code:  26649 
Deposited On:  15. Oct 2020 12:12 
Last Modified:  23. Oct 2020 13:41 