Contraction: A Unified Perspective of Correlation Decay and Zero-Freeness of 2-Spin Systems

Shao, Shuai; Sun, Yuxin

doi:10.1007/s10955-021-02831-0

Cited by 17 publications

(20 citation statements)

References 49 publications

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Section: Introductionmentioning

confidence: 99%

“…Understanding the connection between strong spatial mixing and absence of zeros on families of graphs like G ∆ has recently started to receive attention [20,27,28,36]. In particular in [27,36] it is shown that a standard method for proving strong spatial mixing can be used to prove absence of zeros for partition functions of several models. Very recently, Gamarnik [20] showed that absence of zeros of the partition function implies a weaker form of strong spatial mixing, but his result does not apply to all bounded degree graphs.…”

Section: Introductionmentioning

confidence: 99%

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Absence of zeros implies strong spatial mixing

Regts¹

2021

Preprint

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In this paper we show that absence of complex zeros of the partition function of the hard-core model on any family of bounded degree graphs implies that the associated probability measure, the hard-core measure, satisfies strong spatial mixing on that family. As a corollary we obtain that the hard-core measure on the family of bounded degree claw-free graphs satisfies strong spatial mixing. We furthermore derive strong spatial mixing for graph homomorphism measures from absence of zeros of the graph homomorphism partition function.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Absence of zeros implies strong spatial mixing

Regts¹

2021

Preprint

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“…And if ( , , ) lies outside this regime, the sampling problem becomes computationally intractable [SS12,GŠV15]. Similar bounds were also achieved by another family of critical algorithms based on polynomial interpolation approach for approximating non-vanishing polynomials [PR19, LSS19,SS20]. A glaring issue with both these families of critical algorithms is the expensive time cost, usually in a form of (log Δ) , which is due to enumerating (log )-sized local structures in these algorithms.…”

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confidence: 75%

Rapid mixing of Glauber dynamics via spectral independence for all degrees

Chen¹,

Feng²,

Yin³

et al. 2021

Preprint

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A. We prove an optimal Ω −1 lower bound on the spectral gap of Glauber dynamics for antiferromagnetic two-spin systems with vertices in the tree uniqueness regime. This spectral gap holds for all, including unbounded, maximum degree Δ. Consequently, we have the following mixing time bounds for the models satisfying the uniqueness condition with a slack ∈ (0, 1):1.2. Results for spectrally independent joint distributions. Let be a set of Boolean random variables, and a distribution over {−1, +1} . We use Ω( ) to denote the support of . A configuration ∈ {−1, +1} is feasible if ∈ Ω( ). For any subset Λ ⊆ , let Λ denote the marginal distribution on Λ projected from . A partial configuration ∈ {−1, +1} Λ , where Λ ⊆ , is feasible if ∈ Ω( Λ ). For Λ ∈ Ω( Λ ), we use Λ to denote the distribution over {−1, +1} induced by conditioned on the configuration on Λ being fixed as Λ . Formally:

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“…In particular, many of the recent advances on the development of approximation algorithms for counting problems have been based on viewing the partition function as a polynomial of the underlying parameters in the complex plane, and using refined interpolation techniques from [1,32] to obtain efficient approximation schemes, even for real values [17,18,27,4,2,28,36,35,34]. The bottleneck of this approach is establishing zero-free regions in the complex plane of the polynomials, which in turn requires an in-depth understanding of the models with complex-valued parameters.…”

Section: Introductionmentioning

confidence: 99%

Lee-Yang zeros and the complexity of the ferromagnetic Ising Model on bounded-degree graphs

Buys¹,

Galanis

Patel

et al. 2021

Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA)

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We study the computational complexity of approximating the partition function of the ferromagnetic Ising model in the Lee-Yang circle of zeros given by |λ| = 1, where λ is the external field of the model.Complex-valued parameters for the Ising model are relevant for quantum circuit computations and phase transitions in statistical physics, but have also been key in the recent deterministic approximation scheme for all |λ| = 1 by Liu, Sinclair, and Srivastava. Here, we focus on the unresolved complexity picture on the unit circle, and on the tantalising question of what happens in the circular arc around λ = 1, where on one hand the classical algorithm of Jerrum and Sinclair gives a randomised approximation scheme on the real axis suggesting tractability, and on the other hand the presence of Lee-Yang zeros alludes to computational hardness.Our main result establishes a sharp computational transition at the point λ = 1; in fact, our techniques apply more generally to the whole unit circle |λ| = 1. We show #P-hardness for approximating the partition function on graphs of maximum degree ∆ when b, the edge-interaction parameter, is in the interval (0, ∆−2 ∆ ] and λ is a nonreal on the unit circle. This result contrasts with known approximation algorithms when |λ| = 1 or b ∈ ( ∆−2 ∆ , 1), and shows that the Lee-Yang circle of zeros is computationally intractable, even on bounded-degree graphs.Our inapproximability result is based on constructing rooted tree gadgets via a detailed understanding of the underlying dynamical systems, which are further parameterised by the degree of the root. The ferromagnetic Ising model has radically different behaviour than previously considered antiferromagnetic models, and showing our #P-hardness results in the whole Lee-Yang circle requires a new high-level strategy to construct the gadgets. To this end, we devise an elaborate inductive procedure to construct the required gadgets by taking into account the dependence between the degree of the root of the tree and the magnitude of the derivative at the fixpoint of the corresponding dynamical system.The full version (with all proofs) is available on arXiv at arxiv.org/abs/2006.14828.

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Contraction: A Unified Perspective of Correlation Decay and Zero-Freeness of 2-Spin Systems

Cited by 17 publications

References 49 publications

Absence of zeros implies strong spatial mixing

Absence of zeros implies strong spatial mixing

Rapid mixing of Glauber dynamics via spectral independence for all degrees

Lee-Yang zeros and the complexity of the ferromagnetic Ising Model on bounded-degree graphs

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