Research on replica symmetry breaking in sparsely connected spin glass models

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The objective of this research is to explore the replica symmetry breaking (RSB) phenomena observed in sparsely connected spin glass models of Ising spins. In particular, we seek for a general expression of the de Almeida-Thouless (AT) condition which signals the local instability for RSB and examine properties of the RSB phase of the sparsely-connected models. The obtained results are summarized as follows.1. A general expression of the AT condition was obtained by evaluation of the spin glass susceptibility utilizing the belief propagation algorithm. We also found that the AT condition is determined by a stochastic process that is subject to a large deviation statistics, and developed a methodology to assess the large deviation rate function that characterizes the stochastic process.2. An algorithm to efficiently search Bethe approximation solutions is developed. We performed the Hessian analysis of the Bethe free energy using this algorithm and found that an expected finite size scaling relation is likely to hold for solutions of the Bethe approximation when the external field is relatively small although numerical determination of the critical point is difficult.3. For Markov Chain Monte Carlo-based assessment of the critical point, we examined properties of three measures, the principal eigen value of the susceptibility matrix A, that of the spin glass susceptibility matrix ASG and the spin glass susceptibility XSG. We found that for A and ASG, certain bias corrections are indispensable due to finite-size effect although ASG is most preferred in terms of accuracy of data.4. Properties in the limit of vanishing temperature was examined by using the replica method. The analysis indicates that the ground states are characterized by the one step RSB solution. This property is in contrast with that of the fully connected models, the zero temperature states of which are generally described by the full RSB solution.