On zero-free regions for the anti-ferromagnetic Potts model on bounded-degree graphs

Bencs, Ferenc; Davies, Ewan; Patel, Viresh; Regts, Guus

doi:10.4171/aihpd/108

Cited by 11 publications

(13 citation statements)

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“…On the other side, much less is known about the existence of efficient algorithms for approximating Z(G; q, w) or sampling from the measure Pr G;q,w for the class of bounded degree graphs when w > w c . Implicit in [BDPR21] there is an efficient algorithm for this problem whenever 1 ≥ w > 1 − αq/∆, with α = 1/e, which has been improved to α = 1/2 in [LSS19].…”

Section: Motivation From Computer Sciencementioning

confidence: 99%

Uniqueness of the Gibbs measure for the anti-ferromagnetic Potts model on the infinite $Δ$-regular tree for large $Δ$

Bencs¹,

Boer²,

Buys³

et al. 2022

Preprint

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Section: Motivation From Computer Sciencementioning

confidence: 99%

Uniqueness of the Gibbs measure for the anti-ferromagnetic Potts model on the infinite $Δ$-regular tree for large $Δ$

Bencs¹,

Boer²,

Buys³

et al. 2022

Preprint

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“…The second inequality of this derivation uses the assumed equality in (5). This yields the desired contradiction.…”

Section: Complex Dynamics Preliminaries and The Inductive Step In The...mentioning

confidence: 86%

“…In particular, recent advances on the development of approximation algorithms for counting problems have largely been based on viewing the partition function as a polynomial of the underlying parameters in the complex plane, and using refined interpolation techniques from [1,37] to obtain efficient approximation schemes, even for real values [21,22,31,5,2,32,42,40,39]. 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%

“…Note in particular that all fields implemented by trees lie on the unit circle S, cf. footnote (5). The following theorem, which lies at the heart of the construction of the gadgets, asserts that throughout the relevant range of the parameters we can in fact implement a field arbitrarily close to any number in S. We use T d+1 to denote the set of all rooted trees with maximum degree ≤ d + 1 whose roots have degree ≤ d.…”

Section: Introductionmentioning

confidence: 99%

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

Buys¹,

Galanis²,

Patel³

et al. 2020

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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 edgeinteraction parameter, is in the interval (0, ∆−2 ∆ ] and λ is a non-real 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.Date: July 6, 2020. 1 The parametrisation of the Ising model in terms of δ(σ) follows the closely related works [33, 40]; if instead the model is defined in terms of the number of edges with the same spins the edge-interaction parameter 1/b ∈ (1, ∞) is obtained, whose logarithm is known as the inverse temperature in the physics literature.

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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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On zero-free regions for the anti-ferromagnetic Potts model on bounded-degree graphs

Cited by 11 publications

References 42 publications

Uniqueness of the Gibbs measure for the anti-ferromagnetic Potts model on the infinite $Δ$-regular tree for large $Δ$

Uniqueness of the Gibbs measure for the anti-ferromagnetic Potts model on the infinite $Δ$-regular tree for large $Δ$

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

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

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