Persistent Link:
http://hdl.handle.net/10150/195253
Title:
Spectral Properties of the Renormalization Group
Author:
Yin, Mei
Issue Date:
2010
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
The renormalization group (RG) approach is largely responsible for the considerable success which has been achieved in developing a quantitative theory of phase transitions. This work investigates various spectral properties of the RG map for Ising-type classical lattice systems. It consists of four parts. The first part carries out some explicit calculations of the spectrum of the linearization of the RG at infinite temperature, and discovers that it is of an unusual kind: dense point spectrum for which the adjoint operators have no point spectrum at all, but only residual spectrum. The second part presents a rigorous justification of the existence and differentiability of the RG map in the infinite volume limit at high temperature by a cluster expansion approach. The third part continues the theme of the third part, and shows that the matrix of partial derivatives of the RG map displays an approximate band property for finite-range and translation-invariant Hamiltonians at high temperature. The last part justifies the differentiability of the RG map in the infinite volume limit at the critical temperature under a certain condition. In summary, the first part deals with special cases where exact computations can be done, whereas the remaining parts are concerned with a general theory and provide a mathematically sound base.
Type:
text; Electronic Dissertation
Keywords:
cluster expansion; linearization; renormalization group
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Mathematics; Graduate College
Degree Grantor:
University of Arizona
Advisor:
Faris, William G.
Committee Chair:
Faris, William G.

Full metadata record

DC FieldValue Language
dc.language.isoENen_US
dc.titleSpectral Properties of the Renormalization Groupen_US
dc.creatorYin, Meien_US
dc.contributor.authorYin, Meien_US
dc.date.issued2010en_US
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.description.abstractThe renormalization group (RG) approach is largely responsible for the considerable success which has been achieved in developing a quantitative theory of phase transitions. This work investigates various spectral properties of the RG map for Ising-type classical lattice systems. It consists of four parts. The first part carries out some explicit calculations of the spectrum of the linearization of the RG at infinite temperature, and discovers that it is of an unusual kind: dense point spectrum for which the adjoint operators have no point spectrum at all, but only residual spectrum. The second part presents a rigorous justification of the existence and differentiability of the RG map in the infinite volume limit at high temperature by a cluster expansion approach. The third part continues the theme of the third part, and shows that the matrix of partial derivatives of the RG map displays an approximate band property for finite-range and translation-invariant Hamiltonians at high temperature. The last part justifies the differentiability of the RG map in the infinite volume limit at the critical temperature under a certain condition. In summary, the first part deals with special cases where exact computations can be done, whereas the remaining parts are concerned with a general theory and provide a mathematically sound base.en_US
dc.typetexten_US
dc.typeElectronic Dissertationen_US
dc.subjectcluster expansionen_US
dc.subjectlinearizationen_US
dc.subjectrenormalization groupen_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineMathematicsen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.advisorFaris, William G.en_US
dc.contributor.chairFaris, William G.en_US
dc.contributor.committeememberKennedy, Thomasen_US
dc.contributor.committeememberPickrell, Douglasen_US
dc.contributor.committeememberSims, Roberten_US
dc.identifier.proquest11127en_US
dc.identifier.oclc752260985en_US
All Items in UA Campus Repository are protected by copyright, with all rights reserved, unless otherwise indicated.