The objective of this thesis is to study the structure of the Gibbs measure and sharp phase transitions for mean field spin glasses. In the first part, we study the structure of Gibbs measures in these systems at finite particle number. We prove that, in an appropriate sense, the predicted structure of the infinite volume Gibbs measure for these system arises at large but finite particle numbers. We use this result to settle some related conjectures.
In the second part of this thesis, we study sharp phase transitions. We begin by studying mixed p-spin glasses with Ising spins. We develop an elementary, PDE and Stochastic analysis based approach to the study of the Parisi functional. As an application, we provide an elementary proof of the strict convexity of this functional. We then use this approach to prove sharp results regarding the Replica Symmetric to Replica Symmetry Breaking (RSB) phase transition. In particular, we study a generalization of the conjectured phase boundary of de Almeida and Thouless to all mixed p-spin glasses and prove that it is correct in all models in the temperature-external field plane less a bounded set.
We then turn to the study of mixed p-spin glasses on the sphere. We focus on the low temperature asymptotics of this model. To this end, we identify the Gamma-limit of the Crisanti-Sommers variational problem, thereby establishing a rigorous formula for the ground state energy and moderate deviations for overlap distributions. We study sharp transitions in this problem through the analysis of its dual, which is shown to be an obstacle-type problem. Through this duality, we find a correspondence between the nature of the RSB phase in the primal problem and topological properties of the contact set of the dual. Finally, using this dual approach, we unify two independent conjectures in the mathematics and physics literature. We then analyze the conjectures.
|Advisor:||Ben Arous, Gerard|
|Commitee:||Austin, Tim D., Kohn, Robert V., Newman, Charles M., Varadhan, S. R. S.|
|School:||New York University|
|School Location:||United States -- New York|
|Source:||DAI-B 78/01(E), Dissertation Abstracts International|
|Keywords:||Calculus of variations, Disordered media, Probability, Sherrington-Kirkpatrick, Spin glasses, Statistical physics|
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