A Random Hierarchical Laplacian

1 online resource (71 pages) : PDF

2013

University of North Carolina at Charlotte

The Hierarchical Laplacian, proposed in Dyson's theory of one-dimensional ferromagnetic phase transitions, has a discrete spectrum with each isolated eigenvalue having infinite multiplicity. As a result, the integrated density of states is piecewise constant and the density of states is a sum of point-masses located on its spectrum. To correct these "defects," we modify the Hierarchical Laplacian by allowing its deterministic coefficients to instead vary randomly, but without changing the eigenfunctions. The resulting spectrum is deterministic but the eigenvalues are now random with finite multiplicity and we obtain an absolutely continuous density of states. Examining the eigenvalue statistics near an individual point of the spectrum, we find that, locally, the spectrum is approximately a Poisson point process.

doctoral dissertations

Mathematics

Ph.D.

Density of States
Eigenvalue Statistics
Hierarchical Laplacian
Hierarchical Lattice
Hierarchical Random Walk
Poisson Statistics

Applied Mathematics

Molchanov, Stanislav

Vainberg, Boris
Godin, Yuri
Cherukuri, Harish

Thesis (Ph.D.)--University of North Carolina at Charlotte, 2013.

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Ray_uncc_0694D_10549

http://hdl.handle.net/20.500.13093/etd:1667