Zeros of ferromagnetic 2-spin systems

Guo, Heng; Liu, Jingcheng; Lu, Pinyan

doi:10.1137/1.9781611975994.11

Cited by 15 publications

(20 citation statements)

References 17 publications

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“…In this paper, we look at the computational complexity and complex zeros of the partition function in the Ising model. This is a classical and also currently very active area of research; see [6], [9], [10], [13], [16], [14], [15], [18], [19] and [22] for some recent results.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…From Theorem 1, the by-now-standard polynomial interpolation argument (see [3], [9], [14], [15], [17]) produces an algorithm for approximating the partition function S when the coefficients and are real and satisfy the condition :…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In the case of a quadratic polynomial, the bound (1) states that the Lipschitz constant of the non-linear part of : {−1, 1} −→ R does not exceed 1 − with respect to the ℓ 1 metric (which is twice the Hamming metric) on the Boolean cube {−1, 1} . This is where our approach to approximation differs from those of [9], [13], [14], [15], [19] and [22], which require a uniform bound on the strength of individual interactions , such as max…”

Section: What's Newmentioning

confidence: 99%

“…A novelty of our approach with respect to locating zero-free regions of S , compared to those of [9], [14], [6] and [18], is that we allow all parameters and to vary: this concerns both Lee-Yang zeros [12], [6], [18] of S as a function of with fixed and Fisher zeros [14] of S as a function of with fixed. It appears that Theorem 1 is the first result establishing an asymptotically optimal zero-free region when the interactions are allowed to differ for different pairs { , } and even to be of different signs, so that we have a mixture of ferromagnetic and antiferromagnetic interactions.…”

Section: What's Newmentioning

confidence: 99%

“…Since in our case deg ≤ 4 , to approximate S = (1) within relative error 0 < < 1, it suffices to compute (0) for = ln − ln , where the implied constant in the ' ' notation depends on alone. From (9), we obtain…”

Section: Computing ( ) (0)mentioning

confidence: 99%

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More on zeros and approximation of the Ising partition function

Barvinok

2021

Forum of Mathematics, Sigma

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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.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: What's Newmentioning

confidence: 99%

Section: What's Newmentioning

confidence: 99%

Section: Computing ( ) (0)mentioning

confidence: 99%

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More on zeros and approximation of the Ising partition function

Barvinok

2021

Forum of Mathematics, Sigma

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Mechanism Design with Predictions for Facility Location Games with Candidate Locations

Fang,

Liu

et al. 2024

Lecture Notes in Computer Science

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The complexity of approximating the complex-valued Potts model

2022

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We study the complexity of approximating the partition function of the q-state Potts model and the closely related Tutte polynomial for complex values of the underlying parameters. Apart from the classical connections with quantum computing and phase transitions in statistical physics, recent work in approximate counting has shown that the behaviour in the complex plane, and more precisely the location of zeros, is strongly connected with the complexity of the approximation problem, even for positive real-valued parameters. Previous work in the complex plane by Goldberg and Guo focused on q = 2, which corresponds to the case of the Ising model; for q > 2, the behaviour in the complex plane is not as well understood and most work applies only to the real-valued Tutte plane. Our main result is a complete classification of the complexity of the approximation problems for all non-real values of the parameters, by establishing #P-hardness results that apply even when restricted to planar graphs. Our techniques apply to all q $$\geq$$ ≥ 2 and further complement/refine previous results both for the Ising model and the Tutte plane, answering in particular a question raised by Bordewich, Freedman, Lovász and Welsh in the context of quantum computations.

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Zeros of ferromagnetic 2-spin systems

Cited by 15 publications

References 17 publications

More on zeros and approximation of the Ising partition function

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