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 $f:\{-1,1{\}}^n⟶\mathbb{R}$ 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.

Several improvements

References in corpus (9)

• Inapproximability of the Partition Function for the Antiferromagnetic Ising and Hard-Core Models
• Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials
• Fisher zeros and correlation decay in the Ising model
• The Ising Partition Function: Zeros and Deterministic Approximation
• Location of zeros for the partition function of the Ising model on bounded degree graphs
• Correlation Decay up to Uniqueness in Spin Systems
• Lee-Yang Zeros of the antiferromagnetic Ising Model
• FPTAS for Hardcore and Ising Models on Hypergraphs
• Approximating real-rooted and stable polynomials, with combinatorial applications

Cited by in corpus (1)

• Spectral Independence via Stability and Applications to Holant-Type Problems