More on zeros and approximation of the Ising partition function
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We consider the problem of computing ∑_x e^f(x), where f(x)=∑_ij a_ijξ_i ξ_j + ∑_i b_i ξ_i is a real-valued quadratic function and x=(ξ_1, ..., ξ_n) ranges over the Boolean cube {-1, 1}^n. We prove that for any δ >0, fixed in advance, the value of ∑_x e^f(x) can be approximated within relative error 0 < ϵ < 1 is quasi-polynomial n^O(ln n - lnϵ) time, as long as ∑_j |a_ij| ≤ 1-δ for all i. We apply the method of polynomial interpolation, for which we prove that ∑_x e^f(x) 0 for complex a_ij and b_i such that ∑_j | a_ij| ≤ 1-δ, ∑_j | a_ij| ≤δ^2/10 and | b_i| ≤δ^2/10 for all i, which is interpreted as the absence of a phase transition in the Lee - Yang sense in the corresponding Ising model. The bounds are asymptotically optimal. The novel feature of the bounds is that they control the total interaction of each vertex but not every pairwise interaction.
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