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On the p, q-binomial distribution and the Ising model
2010 (English)In: PHILOSOPHICAL MAGAZINE, Vol. 90, no 24, 3313-3353 p., PII 923766914Article in journal (Refereed) Published
Abstract [en]

We employ p, q-binomial coefficients, a generalisation of the binomial coefficients, to describe the magnetisation distributions of the Ising model. For the complete graph this distribution corresponds exactly to the limit case p = q. We apply our investigation to the simple d-dimensional lattices for d = 1, 2, 3, 4, 5 and fit p, q-binomial distributions to our data, some of which are exact but most are sampled. For d = 1 and d = 5, the magnetisation distributions are remarkably well-fitted by p,q-binomial distributions. For d = 4 we are only slightly less successful, while for d = 2, 3 we see some deviations (with exceptions!) between the p, q-binomial and the Ising distribution. However, at certain temperatures near Tc the statistical moments of the fitted distribution agree with the moments of the sampled data within the precision of sampling. We begin the paper by giving results of the behaviour of the p, q-distribution and its moment growth exponents given a certain parameterisation of p, q. Since the moment exponents are known for the Ising model (or at least approximately for d = 3) we can predict how p, q should behave and compare this to our measured p, q. The results speak in favour of the p, q-binomial distribution's correctness regarding its general behaviour in comparison to the Ising model. The full extent to which they correctly model the Ising distribution, however, is not settled.

Place, publisher, year, edition, pages
2010. Vol. 90, no 24, 3313-3353 p., PII 923766914
National Category
Discrete Mathematics
URN: urn:nbn:se:umu:diva-107766DOI: 10.1080/14786435.2010.484406ISI: 000279449000003OAI: diva2:851491
Available from: 2015-09-05 Created: 2015-08-28 Last updated: 2015-09-07

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Lundow, P. H.
Discrete Mathematics

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