Solvable Particle Models Related to the Beta-Ensemble
| dc.contributor.advisor | Sinclair, Christopher | en_US |
| dc.contributor.author | Shum, Christopher | en_US |
| dc.date.accessioned | 2013-10-03T23:35:27Z | |
| dc.date.available | 2013-10-03T23:35:27Z | |
| dc.date.issued | 2013-10-03 | |
| dc.description.abstract | For beta > 0, the beta-ensemble corresponds to the joint probability density on the real line proportional to prod_{n > m}^N abs{x_n - x_m}^beta prod_{n = 1}^N w(x_n) where w is the weight of the system. It has the application of being the Boltzmann factor for the configuration of N charge-one particles interacting logarithmically on an infinite wire inside an external field Q = -log w at inverse temperature beta. Similarly, the circular beta-ensemble has joint probability density proportional to prod_{n > m}^N abs{e^{itheta_n} - e^{itheta_m}}^beta prod_{n = 1}^N w(x_n) quad for theta_n in [- pi, pi) and can be interpreted as N charge-one particles on the unit circle interacting logarithmically with no external field. When beta = 1, 2, and 4, both ensembles are said to be solvable in that their correlation functions can be expressed in a form which allows for asymptotic calculations. It is not known, however, whether the general beta-ensemble is solvable. We present four families of particle models which are solvable point processes related to the beta-ensemble. Two of the examples interpolate between the circular beta-ensembles for beta = 1, 2, and 4. These give alternate ways of connecting the classical beta-ensembles besides simply changing the values of beta. The other two examples are "mirrored" particle models, where each particle has a paired particle reflected about some point or axis of symmetry. | en_US |
| dc.identifier.uri | https://hdl.handle.net/1794/13302 | |
| dc.language.iso | en_US | en_US |
| dc.publisher | University of Oregon | en_US |
| dc.rights | All Rights Reserved. | en_US |
| dc.subject | Beta Ensemble | en_US |
| dc.subject | Random Matrix Theory | en_US |
| dc.title | Solvable Particle Models Related to the Beta-Ensemble | en_US |
| dc.type | Electronic Thesis or Dissertation | en_US |
| thesis.degree.discipline | Department of Mathematics | en_US |
| thesis.degree.grantor | University of Oregon | en_US |
| thesis.degree.level | doctoral | en_US |
| thesis.degree.name | Ph.D. | en_US |
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