Replica symmetry breaking in the `small world' spin glass

Abstract

We apply the cavity method to a spin glass model on a `small world' lattice, a random bond graph super-imposed upon a 1-dimensional ferromagnetic ring. We show the correspondence with a replicated transfer matrix approach, up to the level of one step replica symmetry breaking (1RSB). Using the scheme developed by Mézard & Parisi for the Bethe lattice, we evaluate observables for a model with fixed connectivity and $\pm J$ long range bonds. Our results agree with numerical simulations significantly better than the replica symmetric (RS) theory.

21 pages, 3 figures

References in corpus (5)

• Parallel dynamics of disordered Ising spin systems on finitely connected random graphs
• Replicated Transfer Matrix Analysis of Ising Spin Models on `Small World' Lattices
• Analytic solution of attractor neural networks on scale-free graphs
• Critical Properties of an Ising Model with Dilute Long-range Interactions
• Diagonalization of replicated transfer matrices for disordered Ising spin systems

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• Boolean decision problems with competing interactions on scale-free networks: Critical thermodynamics
• Spin-glass behaviour on random lattices
• Exact results for the Ising model on a small-world network
• Statistical mechanics of LDPC codes on channels with memory
• Spin-glass transition in the Ising model on multiplex networks