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Studies in Statistical Mechanics and Supersymmetric Lattice Models https://doi.org/10.17863/CAM.117521 Wildemann, Peter Bauerschmidt, Roland This work concerns an array of probabilistic models in the context of lattice systems and statistical field theory. Firstly, motivated by predictions about the Anderson transition, we study two distinct but related models on regular tree graphs: the vertex-reinforced jump process (VRJP), a random walk that prefers to jump to previously visited sites, and the $\mathbb{H}^{2|2}$-model, a lattice spin system whose spins take values in a supersymmetric extension of the hyperbolic plane. Both models undergo a phase transition, and our work provides detailed information about the supercritical phase up to the critical point. Moreover, we consider the rigorous construction of the Schwarzian field theory, a measure on the quotient $\mathrm{Diff}(S^{1})/\mathrm{PSL}(2,\RR)$ of circle diffeomorphisms, which has gained popularity in recent theoretical physics literature. Its partition function is calculated by the rigorous implementation of an argument by Belokurov and Shavgulidze. This method exploits a regularisation of the measure, motivated by the theory of Virasoro coadjoint orbits. We also provide motivation for the physical origins of the Schwarzian field theory and offer background on the theory of coadjoint orbits. Furthermore, we consider the graphical representations of the Ising model, including the random cluster, loop $\mathrm{O}(1)$, and random current model. Considering these models as percolation-type random graph models in their own right, we are interested in their monotonicity behaviour. We construct some tree-like graphs for which the loop $\mathrm{O}(1)$ and random current model exhibit a non-unique phase transition. As a consequence there exist infinite graphs $\mathbb{G}\subseteq \mathbb{G}'$ such that the uniform even subgraph of $\mathbb{G}'$ percolates and the uniform even subgraph of $\mathbb{G}$ does not. Moreover, we show that in general the percolation thresholds of the models do not agree. University of Cambridge 2024-12-18 Thesis Doctoral Doctor of Philosophy (PhD) eng https://www.repository.cam.ac.uk/handle/1810/382939 https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/54847c0f-26b5-490a-9e02-c6008ee666bf/download cd87734e15c5b7acccb04b667be243f1 https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/ac5fc480-2976-4b07-9f05-6cd3364a0453/download 87eda9de84448d1f82354d60eee3eb5f https://creativecommons.org/licenses/by/4.0/ lattice model probability reinforced random walk schwarzian field theory spin system statistical mechanics supersymmetry