More on zeros and approximation of the Ising partition function

Abstract

We consider the problem of computing the partition function $\sum _x e^{f(x)}$ , where $f: \{-1, 1\}^n \longrightarrow {\mathbb R}$ 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 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 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.

Smoothed counting of 0–1 points in polyhedra

“…In [6] we prove that for any 0 < 𝛿 < 1, fixed in advance, the value of (4.1) can be approximated within relative error 0 < 𝜖 < 1 in quasi-polynomial n O 𝛿 (ln n−ln 𝜖) time provided…”

“…Other than that, the complexity does not depend on f j . The result of [6] allows us to handle certain sparse systems (1.1). Namely, let us fix integers r i ≥ 1 for i = 1, … , m, integer c ≥ 1 and suppose that the matrix A = ( 𝛼 ij ) contains at most r i non-zero entries in the ith row and at most c non-zero entries in each column, while all entries satisfy the inequalities…”

“…Note that there is no dependence on the right hand sides 𝛽 i or probabilities p j . We note that the system of equations (3.1) for perfect matchings in a k-uniform Δ-regular hypergraph is not sparse in the above sense when k is fixed but Δ is allowed to grow, and Theorem 1.2 does not follow from the results of [6].…”

“…In [6] we prove that for any $$$ 0<\delta <1 $$$, fixed in advance, the value of () can be approximated within relative error $$$ 0<\epsilon <1 $$$ in quasi‐polynomial $$$ {n}^{O_{\delta}\left(\ln n-\ln \epsilon \right)} $$$ time provided $$$$ \sum \limits_{j:\kern0.3em j\ne k}\left|{g}_{jk}\right|\le 1-\delta \kern1em \mathrm{for}\kern1em k=1,\dots, n. $$$$ Geometrically, the condition () means that the Lipschitz constant of the quadratic form $$$$ \sum \limits_{1\le k<j\le n}{g}_{kj}{\eta}_k{\eta}_j, $$$$ on the Boolean cube $$$ {\left\{-1,1\right\}}^n $$$ endowed with the $$$ {\ell}^1 $$$‐metric does not exceed $$$ 1-\delta $$$ (the condition is essentially sharp, modulo NP $$$ \ne $$$ BPP hypothesis). To avoid dealing with exponentially large numbers, we assume that the coefficients …”

“…The result of [6] allows us to handle certain sparse systems (). Namely, let us fix integers $$$ {r}_i\ge 1 $$$ for $$$ i=1,\dots, m $$$, integer $$$ c\ge 1 $$$ and suppose that the matrix $$$ A=\left({\alpha}_{ij}\right) $$$ contains at most $$$ {r}_i $$$ non‐zero entries in the $$$ i $$$th row and at most $$$ c $$$ non‐zero entries in each column, while all entries satisfy the inequalities $$$$ \mid {\alpha}_{ij}\mid \le 1\kern1em \mathrm{for}\ \mathrm{all}\kern1em i,j.$$…”