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Title: Mathematical aspects of some mean field spin glass models
Author: Wedagedera, J. R.
Awarding Body: University of Wales Swansea
Current Institution: Swansea University
Date of Award: 1998
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In Chapter 2 we review the first theoretical model for spin glasses proposed by Edwards and Anderson [25, 26], the Sherrington Kirkpatrick Model [69, 70], Parisi's heuristic scheme [57] proposed to solve the SK model and Derrida's Random Energy Model (REM) [9, 10]. In Chapter 3 we state some well known theorems from the main probabilistic tool that we use, namely the Large Deviation Theory (LDT) and also some related convexity results. The free energies of the mean-field Ising model and the REM are rigorously derived as examples. In Chapter 4 a separable spin glass model which was solved by van Hemmen et al [78] using LDT, is rigorously treated. The almost sure convergence criteria associated with the cumulant generating function C(t) with respect to the quenched random ξ is carefully investigated and the free-energy is re-derived using LD arguments. This work has been accepted for publication in the Journal of Applied Mathematics and Stochastic Analysis [22]. The solutions of the Ising model and an Ising spin glass model on Cayley trees are discussed in Chapter 5. The directed polymer problem on Cayley trees and its solution by Derrida and Spohn [14] via the Generalized REM is also discussed. In Chapter 6 we solve rigorously a spin glass problem on a Cayley tree with higher-order ferromagnetic interactions. Using a level-I large deviation argument together with the martingale approach used by Buffet, Patrick and Pulé [3], explicit expressions for the free energy are derived in different regions of the phase diagram. Sourlas [71] discovered a connection between the REM and the error correcting codes used in telecommunications. The results obtained in Chapter 3 and Chapter 6 are being used in investigate this idea in Chapter 7. In Chapter 8 we discuss a computer simulation which uses the method of coincidence counting [49], to compute the entropy of the spin glass model which we treated in Chapter 5.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available