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Aumann, Simon (2014): Nearcritical percolation and crystallisation. Dissertation, LMU München: Faculty of Mathematics, Computer Science and Statistics



This thesis contains results on singularity of nearcritical percolation scaling limits, on a rigidity estimate and on spontaneous rotational symmetry breaking. First it is shown that - on the triangular lattice - the laws of scaling limits of nearcritical percolation exploration paths with different parameters are singular with respect to each other. This generalises a result of Nolin and Werner, using a similar technique. As a corollary, the singularity can even be detected from an infinitesimal initial segment. Moreover, nearcritical scaling limits of exploration paths are mutually singular under scaling maps. Second full scaling limits of planar nearcritical percolation are investigated in the Quad-Crossing-Topology introduced by Schramm and Smirnov. It is shown that two nearcritical scaling limits with different parameters are singular with respect to each other. This result holds for percolation models on rather general lattices, including bond percolation on the square lattice and site percolation on the triangular lattice. Third a rigidity estimate for 1-forms with non-vanishing exterior derivative is proven. It generalises a theorem on geometric rigidity of Friesecke, James and Müller. Finally this estimate is used to prove a kind of spontaneous breaking of rotational symmetry for some models of crystals, which allow almost all kinds of defects, including unbounded defects as well as edge, screw and mixed dislocations, i.e. defects with Burgers vectors.