D H Bailey et al 2006 J. Phys. A: Math. Gen. 39 12271 doi:10.1088/0305-4470/39/40/001
Integrals of the Ising class
D H Bailey1, J M Borwein2 and R E Crandall3
Show affiliationsFrom an experimental-mathematical perspective we analyse 'Ising-class' integrals. These are structurally related n-dimensional integrals we call Cn, Dn, En, where Dn is a magnetic susceptibility integral central to the Ising theory of solid-state physics. We first analyse 
We had conjectured—on the basis of extreme-precision numerical quadrature—that Cn has a finite large-n limit, namely C∞ = 2 e−2γ, with γ being the Euler constant. On such a numerological clue we are able to prove the conjecture. We then show that integrals Dn and En both decay exponentially with n, in a certain rigorous sense. While Cn, Dn remain unresolved for n ≥ 5, we were able to conjecture a closed form for E5. Our experimental results involved extreme-precision, multidimensional quadrature on intricate integrands; thus, a highly parallel computation was required.
- PACS
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05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
- MSC
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33F10 Symbolic computation (Gosper and Zeilberger algorithms, etc.) (See also 68W30)
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
- Subjects
- Dates
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Issue 40 (6 October 2006)
Received 2 June 2006, in final form 14 August 2006
Published 19 September 2006
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Integrals of the Ising class
D H Bailey et al 2006 J. Phys. A: Math. Gen. 39 12271
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