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

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Abstract
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 QuadCrossingTopology 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 1forms with nonvanishing 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.
Item Type:  Thesis (Dissertation, LMU Munich) 

Keywords:  nearcritical, percolation, exploration path, full scaling limit, singular, rigidity estimate, crystal, spontaneous symmetry breaking, Burgers vector, arbitrary defects 
Subjects:  600 Natural sciences and mathematics 600 Natural sciences and mathematics > 510 Mathematics 
Faculties:  Faculty of Mathematics, Computer Science and Statistics 
Language:  English 
Date Accepted:  26. November 2014 
1. Referee:  Merkl, Franz 
Persistent Identifier (URN):  urn:nbn:de:bvb:19177436 
MD5 Checksum of the PDFfile:  9530334d90b3c33a502be338cd896a17 
Signature of the printed copy:  0001/UMC 22601 
ID Code:  17743 
Deposited On:  20. Jan 2015 14:32 
Last Modified:  20. Jul 2016 10:37 