On the stability of randomly frustrated systems with finite connectivity

Mottishaw, Peter; Dominicis, C. De

doi:10.1088/0305-4470/20/6/007

Cited by 76 publications

(54 citation statements)

References 7 publications

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“…Despite several efforts, the question of how to handle RSB in finite-connectivity systems is still open. Most attempts [41,42,43,44] try to apply the first step of Parisi's RSB scheme [38] which however was originally developped for infinite-connectivity spin glasses. Due to the more complex structure of the order parameter a complete solution is however still missing.…”

Section: The Simplest One-step Replica Symmetry Broken Solutionmentioning

confidence: 99%

Minimal vertex covers on finite-connectivity random graphs: A hard-sphere lattice-gas picture

2001

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The minimal vertex-cover (or maximal independent-set) problem is studied on random graphs of finite connectivity. Analytical results are obtained by a mapping to a lattice gas of hard spheres of (chemical) radius one, and they are found to be in excellent agreement with numerical simulations. We give a detailed description of the replica-symmetric phase, including the size and the entropy of the minimal vertex covers, and the structure of the unfrozen component which is found to percolate at connectivity c ≃ 1.43. The replica-symmetric solution breaks down at c = e ≃ 2.72. We give a simple one-step replica symmetry broken solution, and discuss the problems in interpretation and generalization of this solution.

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Section: The Simplest One-step Replica Symmetry Broken Solutionmentioning

confidence: 99%

Minimal vertex covers on finite-connectivity random graphs: A hard-sphere lattice-gas picture

2001

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“…Models of finitely connected disordered spin systems have been studied for some 20 years, following the initiating papers [1][2][3][4][5]. In particular due to the unexpectedly rich and varied range of multi-disciplinary applications of finite connectivity replica techniques which emerged subsequently in, for example, spin-glass modelling [6][7][8][9], error correcting codes [10][11][12][13], theoretical computer science [14][15][16][17], recurrent neural networks [18][19][20] and 'small-world' networks [21], this field is presently enjoying a renewed interest and popularity.…”

Section: Introductionmentioning

confidence: 99%

Finitely connected vector spin systems with random matrix interactions

Coolen¹,

Skantzos²,

Castillo³

et al. 2005

J. Phys. A: Math. Gen.

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We use finite connectivity equilibrium replica theory to solve models of finitely connected unit-length vectorial spins, with random pair-interactions which are of the orthogonal matrix type. Finitely connected spin models, although still of a mean-field nature, can be regarded as a convenient level of description in between fully connected and finite-dimensional ones. Since the spins are continuous and the connectivity c remains finite in the thermodynamic limit, the replica-symmetric order parameter is a functional. The general theory is developed for arbitrary values of the dimension d of the spins, and arbitrary choices of the ensemble of random orthogonal matrices. We calculate phase diagrams and the values of moments of the order parameter explicitly for d = 2 (finitely connected XY spins with random chiral interactions) and for d = 3 (finitely connected classical Heisenberg spins with random chiral interactions). Numerical simulations are shown to support our predictions quite satisfactorily.

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“…In this paper we study the as yet unsolved class of symmetrically diluted attractor neural networks with finite connectivity, where the average number c of bonds per neuron is finite, independent of the system size N. For bond-disordered spin systems, the finite connectivity regime (which so far has been addressed only in the context of spin glasses [11,12,13,14,16,17], error correcting codes [18,19] and satisfiability problems [20,21,22,23]) requires order parameter functions, which generalize the replica matrices of [2]. For finite c the replica symmetry breaking theory (RSB) is still under development [16,24,25,26,17,27].…”

Section: Introductionmentioning

confidence: 99%

Finite connectivity attractor neural networks

Wemmenhove¹,

Coolen²

2003

J. Phys. A: Math. Gen.

103

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Abstract. We study a family of diluted attractor neural networks with a finite average number of (symmetric) connections per neuron. As in finite connectivity spin glasses, their equilibrium properties are described by order parameter functions, for which we derive an integral equation in replica symmetric (RS) approximation. A bifurcation analysis of this equation reveals the locations of the paramagnetic to recall and paramagnetic to spin-glass transition lines in the phase diagram. The line separating the retrieval phase from the spin-glass phase is calculated at zero temperature. All phase transitions are found to be continuous.

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On the stability of randomly frustrated systems with finite connectivity

Cited by 76 publications

References 7 publications

Minimal vertex covers on finite-connectivity random graphs: A hard-sphere lattice-gas picture

Minimal vertex covers on finite-connectivity random graphs: A hard-sphere lattice-gas picture

Finitely connected vector spin systems with random matrix interactions

Finite connectivity attractor neural networks

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