We consider the problem of computing  \sum_x e^{f(x)}, where f (x)=\sum_{ij a_{ij} \xi_i \xi_j + \sum_i b_i \xi_i is a real-valued quadratic function and x=}%DIFDELCMD < \left(%%% \xi_1, \ldots, \xi_n%DIFDELCMD < \right) %%% ranges over{\Bbb   the Boolean cube \{-1, 1\}^n. We prove that for any \delta >0, fixed in advance, the value of \sum_x e^{f(x) can be approximated within relative error 0 < \epsilon < 1 is quasi-polynomial n^{O(\ln n - \ln \epsilon)} time , as long as \sum_j |a_{ij| \leq }  1-\delta for all i. We apply the method of polynomial interpolation, for which we prove that \sum_x e^{ f \tilde{f} (x)} \ne 0 for complex a _{ij and b_i such that \sum_j |\Re}%DIFDELCMD < \thinspace %%% a _{ij| \leq 1-\delta, \sum_j |\Im}%DIFDELCMD < \thinspace %%% a_{ij| \leq \delta^2/10 and |\Im}%DIFDELCMD < \thinspace %%% b_i| \leq \delta^2/10 for all i, which is\tilde{f}  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 .
We consider the problem of computing the partition function \sum_x e^{f(x)}, where f a_{ij} \xi_i \xi_j + \sum_i b_i \xi_i is a real-valued quadratic function and x=}%DIFDELCMD < \left(%%% %DIFDELCMD < \right) %%% : \{-1, 1\}^n \longrightarrow{\Bbb R is a quadratic or cubic polynomial on the Boolean cube \{-1, 1\}^n. In the case of a quadratic polynomial f, we show that the partition function can be approximated within relative error 0 < \epsilon < 1 in quasi-polynomial n^{O(\ln n - \ln \epsilon)} time | \leq } if the Lipschitz constant of the non-linear part of f with respect to the \ell^1 metric on the Boolean cube does not exceed 1-\delta , for any \delta >0, fixed in advance. For a cubic polynomial f, we get the same result under a somewhat stronger condition. We apply the method of polynomial interpolation, for which we prove that \sum_x e^{ \tilde{f} (x)} \ne 0 for and b_i such that \sum_j |\Re}%DIFDELCMD < \thinspace %%% | \leq 1-\delta, \sum_j |\Im}%DIFDELCMD < \thinspace %%% | \leq \delta^2/10 and |\Im}%DIFDELCMD < \thinspace %%% complex-valued polynomials\tilde{f} in a neighborhood of a real-valued f satisfying the above mentioned conditions. The bounds are asymptotically optimal. Results on the zero-free region are interpreted as the absence of a phase transition in the Lee - Yang sense in the corresponding Ising model. The novel feature of the bounds is that they control the total interaction of each vertex but not every single interaction of sets of vertices .
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the partition function
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(x)=\sum_{ij
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\xi_1, \ldots, \xi_n
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ranges over
: \{-1, 1\}^n \longrightarrow
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R
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is a quadratic or cubic polynomial on
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We prove that for any \delta >0, fixed in advance, the value of \sum_x e^{f(x)
In the case of a quadratic polynomial f, we show that the partition function
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is
in
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, as long as \sum_j |a_{ij
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if the Lipschitz constant of the non-linear part of f with respect to the \ell^1 metric on the Boolean cube does not exceed
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for all i.
, for any \delta >0, fixed in advance. For a cubic polynomial f, we get the same result under a somewhat stronger condition.
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f
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complex a _{ij
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a _{ij
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a_{ij
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b_i| \leq \delta^2/10 for all i, which is
complex-valued polynomials
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in a neighborhood of a real-valued f satisfying the above mentioned conditions. The bounds are asymptotically optimal. Results on the zero-free region are
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bounds are asymptotically optimal. The
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pairwise interaction
single interaction of sets of vertices