Properties of minimum spanning trees and fractional quantum Hall states
Thomas Jackson
Properties of minimum spanning trees and fractional quantum Hall states Thomas Sundal Jackson 2010 This dissertation consists of work done on two disjoint problems. In the first two chapters I discuss fractal properties of average-case solutions to the random minimal spanning tree (MST) problem: given a graph with costs on the edges, the MST is the spanning tree minimizing the sum of the total cost of the chosen edges. In the random version the costs are quenched random variables. I solve the random MST problem on the Bethe lattice with appropriate boundary conditions and use the results to infer fractal dimensions in the mean-field approximation. I find that connected components of the MST in a window have dimension D = 6, which establishes the upper critical dimension dc = 6. This contradicts a value dc = 8 proposed previously in the literature; I correct the argument that led to this value. I then develop an exact low-density expansion for the random MST on a finite graph and use it to develop an expansion for the MST on critical percolation clusters. I prove this perturbation expansion is renormalizable around dc = 6. Using a renormalization-group approach, I calculate the fractal dimension Dp of paths on the latter MST to first order in e = 6− d for d ≤ 6, with the result Dp ∼ 2− e/7. In the final chapter, I investigate the correspondence between wavefunctions in the fractional quantum Hall effect obtained as blocks of a conformal field theory (CFT) versus those defined as zero-energy eigenstates of projection Hamiltonians, specifically one which forbids three particles to come together in one of two linearly-independent states of relative angular momentum six and all states of lesser relative angular momentum. I construct zero-energy states from amplitudes of superconformal currents using a result due to Simon. The counting of edge excitations of these states agrees with the character formula for the superconformal Kac vacuum module at generic central charge c, which implies this Hamiltonian is gapless for all c. I attempt to obtain a rational theory by projecting out additional states, focusing on the M(3, 8) and tricritical Ising CFTs. Properties of minimum spanning trees and fractional quantum Hall states A DISSERTATION PRESENTED TO THE FACULTY OF THE GRADUATE SCHOOL OF YALE UNIVERSITY IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY BY THOMAS SUNDAL JACKSON DISSERTATION DIRECTOR: NICHOLAS READ
•Jan 1, 2010•1 citation