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 nearest-neighbor 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 out---hence, between the different regimes of $q$, a phase transition occurs. In the second and third contribution, the well-known site percolation model is considered in high dimensions. We derive the lace expansion, an identity for the model's two-point function, to deduce the triangle condition, the infra-red bound, and thus mean-field 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 mean-field behavior in high dimension through a continuum-space adaption of the lace expansion. Moreover, as an example of the implied mean-field behavior, the critical exponent $\gamma$ is proven to exist and (as on the lattice) to take value 1.