Integrals of the Ising Class

From 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.