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# Effect of platykurtic and leptokurtic distributions in the random-field Ising model: Mean-field approach

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Physical Review E (Impact Factor: 2.29). 07/2009; 80(1 Pt 1):011143. DOI: 10.1103/PhysRevE.80.011143 Source: PubMed

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Nuno Crokidakis, Aug 04, 2014 Available from: Data provided are for informational purposes only. Although carefully collected, accuracy cannot be guaranteed. The impact factor represents a rough estimation of the journal's impact factor and does not reflect the actual current impact factor. Publisher conditions are provided by RoMEO. Differing provisions from the publisher's actual policy or licence agreement may be applicable.

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**ABSTRACT:**Using a single functional form which is able to represent several different classes of statistical distributions, we introduce a preliminary study of the ferromagnetic Ising model on the cubic lattices under the influence of non-Gaussian local external magnetic field. Specifically, depending on the value of the tail parameter, $\tau $ ($\tau < 3$), we assign a quenched random field that can be platykurtic (sub-Gaussian) or leptokurtic (fat-tailed) form. For $\tau< 5/3$, such distributions have finite standard deviation and they are either the Student-$t$ ($1< \tau< 5/3$) or the $r$-distribution ($\tau< 1$) extended to all plausible real degrees of freedom with the Gaussian being retrieved in the limit $\tau \rightarrow 1$. Otherwise, the distribution has got the same asymptotic power-law behaviour as the $\alpha$-stable L\'{e}vy distribution with $\alpha = (3 - \tau)/(\tau - 1)$. The uniform distribution is achieved in the limit $\tau \rightarrow \infty$. Our results purport the existence of ferromagnetic order at finite temperatures for all the studied values of $\tau$ with some mean-field predictions surviving in the three-dimensional case. Comment: Report of the first results of an ongoing project. 12 pages and 8 figures. Comments welcome - [Show abstract] [Hide abstract]

**ABSTRACT:**In this paper, we develop a stochastic process rules (SPR) based Markov chain method to calculate the degree distributions of evolving networks. This new approach overcomes two shortcomings of Shi, Chen and Liu’s use of the Markov chain method (Shi et al. 2005 [21]). In addition we show how an SPR-based Markov chain method can be effectively used to calculate degree distributions of random birth-and-death networks, which we believe to be novel. First SPR are introduced to replace traditional evolving rules (TR), making it possible to compute degree distributions in one sample space. Then the SPR-based Markov chain method is introduced and tested by using it to calculate two kinds of evolving network. Finally and most importantly, the SPR-based method is applied to the problem of calculating the degree distributions of random birth-and-death networks.Physica A: Statistical Mechanics and its Applications 06/2012; 391(11):3350–3358. DOI:10.1016/j.physa.2012.01.040 · 1.73 Impact Factor