Fracturing of Optimal Paths in a Random Lattice
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- Institutt for fysikk 
The subject of this thesis is the study of the creation of fault lines in a random lattice, provoked by the successive failure of optimal paths. Using the recently developed Optimal Path Cracked model, we investigate how central characteristics of the successive optimal paths evolve as the lattice breaks down, and how this progression of characteristics depends on the magnitude of disorder imparted on the lattice. We then see how the OPC model, while originally proposed in the context of the shortest path problem, can be generalized to alternate optimal path problems, namely the minimax problem and the widest path problem. It is shown that for a given lattice, the minimax OPC is equal to the the backbone of the shortest OPC. The widest path OPC, although constituting a distinct object on any lattice, is shown to scale with lattice size in the same manner as the minimax OPC and the backbone of the shortest path OPC; with the fundamental process behind it being closely related to the minimax OPC process. Lastly, we explain the connection between the OPC process and a variety of other phenomena which have previously been shown to exhibit similar scaling behavior. We show how the OPC process for the widest path problem can be reduced to the shortest path problem on the dual lattice using the limit of very high disorder, the so-called ultrametric limit, and how an algorithm based on invasion percolation can be used as a quicker method of finding an OPC.