Order-parameter flow in the SK spin glass. I. Replica symmetry

Coolen, Anthony C. C.; Sherrington, David C.

doi:10.1088/0305-4470/27/23/013

Cited by 31 publications

(79 citation statements)

References 27 publications

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“…This is not unexpected [20]. However, as with the original dynamical replica theory, there is scope for increasing the order parameter set [20,21], which should improve its accuracy systematically, albeit at a numerical cost. At present our method requires the convergence of belief propagation.…”

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confidence: 84%

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Dynamical Replica Analysis of Disordered Ising Spin Systems on Finitely Connected Random Graphs

et al. 2005

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We study the dynamics of macroscopic observables such as the magnetization and the energy per degree of freedom in Ising spin models on random graphs of finite connectivity, with random bonds and/or heterogeneous degree distributions. To do so, we generalize existing versions of dynamical replica theory and cavity field techniques to systems with strongly disordered and locally treelike interactions. We illustrate our results via application to, e.g., +/-J spin glasses on random graphs and of the overlap in finite connectivity Sourlas codes. All results are tested against Monte Carlo simulations.

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confidence: 84%

“…Following the procedure outlined for fully connected systems [20], we consider the evolution of two macroscopic observables, the magnetization ms N ÿ1 i s i , and the internal energy es ÿN ÿ1 i<j c ij J ij s i s j . We abbreviate m; e. One easily derives a Kramers-Moyal (KM) expansion for their probability density P t s p t s ÿ s; see, e.g., [22].…”

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confidence: 99%

Dynamical Replica Analysis of Disordered Ising Spin Systems on Finitely Connected Random Graphs

et al. 2005

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“…Spin glass models with hard spins were first treated in [5,6,7]. A powerful generating functional approach was then developed by Coolen and collaborators [8,9], and it forms the basis of our analysis here; see also [10,11] for reviews of the techniques used in both soft and hard spin models. However, dynamical TAP equations to describe the kinetics of order parameters for a specific realization of the disorder have been only derived for the spherical p-spin model [12] and the stationary state of the Ising spin model with asynchronous update dynamics [13].…”

Section: Introductionmentioning

confidence: 99%

Dynamical TAP equations for non-equilibrium Ising spin glasses

Roudi¹,

Hertz²

2011

J. Stat. Mech.

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Abstract. We derive and study dynamical TAP equations for Ising spin glasses obeying both synchronous and asynchronous dynamics using a generating functional approach. The system can have an asymmetric coupling matrix, and the external fields can be time-dependent. In the synchronously updated model, the TAP equations take the form of self consistent equations for magnetizations at time t + 1, given the magnetizations at time t. In the asynchronously updated model, the TAP equations determine the time derivatives of the magnetizations at each time, again via self consistent equations, given the current values of the magnetizations. Numerical simulations suggest that the TAP equations become exact for large systems.

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“…To analyze the above model mathematically we used the Dynamical Replica Theory (DRT), 7) which results in a nonlinear partial differential equation (PDE) for the joint distribution p(f, b, σ) of single-clone variables (f i , b i , σ i ) (i.e. antibody and B-cell concentrations, and clone sensitivities).…”

Section: Previous Resultsmentioning

confidence: 99%

A Large Scale Dynamical System Immune Network Model with Finite Connectivity

Uezu

Kadono

Hatchett

et al. 2006

Prog. Theor. Phys. Suppl.

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We study a model of an idiotypic immune network which was introduced by N. K. Jerne. It is known that in immune systems there generally exist several kinds of immune cells which can recognize any particular antigen. Taking this fact into account and assuming that each cell interacts with only a finite number of other cells, we analyze a large scale immune network via both numerical simulations and statistical mechanical methods, and show that the distribution of the concentrations of antibodies becomes non-trivial for a range of values of the strength of the interaction and the connectivity. §1. IntroductionThere are many numerical and theoretical studies of biological networks in which there is a global coupling between the constituent elements, e.g. Hopfield-type neural networks and networks of Kuramoto-type phase oscillators. 1), 2) In contrast, relatively few studies have been carried out of networks where each component interacts with only a small number of randomly selected other components. One such system is the immune network, introduced by Jerne 3) to explain the activation of immune cells in the absence of external stimulation.Let us briefly summarize the main mechanism of cell-interaction in the immune system. Its main constituents are B-lymphocytes (B-cells), T-lymphocytes (T-Cells) and antibodies produced by B-cells. B-cells and T-cells have receptors on their surfaces. The receptors of B-cells are antibodies, which recognize and connect to antigens in order to neutralize them; they have specific 3-dimensional structures which are called 'idiotypes'. A family of B-cells which are generated from a given B-cell is called a 'clone'. Hence, all members of a clone as well as the antibodies produced by this clone have the same idiotype. In general, each antibody could present several 3-dimensional structures which can be recognized by other B-cells. This then generates, indirectly, an effective interaction between antibodies. This paper is organized as follows. In §2 we formulate the model, followed by a summary of previous studies in §3. In §4, we present the results of our present study. Section 5 is devoted to a summary and a discussion.

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Order-parameter flow in the SK spin glass. I. Replica symmetry

Cited by 31 publications

References 27 publications

Dynamical Replica Analysis of Disordered Ising Spin Systems on Finitely Connected Random Graphs

Dynamical Replica Analysis of Disordered Ising Spin Systems on Finitely Connected Random Graphs

Dynamical TAP equations for non-equilibrium Ising spin glasses

A Large Scale Dynamical System Immune Network Model with Finite Connectivity

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