Role of the Finite Replica Analysis in the Mean-Field Theory of Spin Glasses

In this thesis, we review and examine the replica method from several viewpoints. The replica method is a mathematical technique to calculate general moments of stochastic variables. This method provides a systematic way to evaluate physical quantities and becomes one of the most important tools in the theory of spin glasses and in the related discipline including information processing tasks. In spite of the effectiveness of the replica method, it is known that several problems exist in the procedures of the method itself. The replica symmetry breaking is the central topic of those problems and is the main issue of this thesis. To elucidate this point, we review the recent progress about the replica symmetry breaking including its physical and mathematical descriptions in detail. Based on those descriptions, several spin-glass models and Ising perceptron are deeply investigated.
Comment: A Thesis submitted in fulfillment of the requirements of Ph.D. in Tokyo Tech. in 2010. 148 pages. Note that evaluation of $y_{AT}$ in Section 4 contains some mistakes (the conclusion is unchanged)