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Author Goldberg, Leslie Ann ♦ Jerrum, Mark ♦ Paterson, Mike
Source CiteSeerX
Content type Text
File Format PDF
Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science
Abstract The subject of this article is spin-systems as studied in statistical physics. We focus on the case of two spins. This case encompasses models of physical interest, such as the classical Ising model (ferromagnetic or antiferromagnetic, with or without an applied magnetic field) and the hard-core gas model. There are three degrees of freedom, corresponding to our parameters , "/ and/. We wish to study the complexity of (approximately) computing the partition function in terms of these parameters. We pay special attention to the symmetric case/ = 1 for which our results are depicted in Figure 1. Exact computation of the partition function Z is NP-hard except in the trivial case fi"/ = 1, so we concentrate on the issue of whether Z can be computed within small relative error in polynomial time. We show that there is a fully polynomial randomised approximation scheme (FPRAS) for the partition function in the "ferromagnetic" region "/ _> 1, but (unless RP = NP) there is no FPRAS in the "antiferromagnetic" region corresponding to the square defined by 0 < < 1 and 0 < ' < 1. Neither of these "natural" regions -- neither the hyperbola nor the square -- marks the boundary between tractable and intractable. In one direction, we provide an FPRAS for the partition function within a region which extends well away from the hyperbola. In the other direction, we exhibit two tiny, symmetric, intractable regions extending beyond the antiferromagnetic region. We also extend our results to the asymmetric case/ 1.
Educational Role Student ♦ Teacher
Age Range above 22 year
Educational Use Research
Education Level UG and PG ♦ Career/Technical Study
Learning Resource Type Article