On a Conjecture of Sokal Concerning Roots of the Independence Polynomial

Peters, Hans; Regts, Guus

doi:10.1307/mmj/1541667626

Cited by 77 publications

(128 citation statements)

References 25 publications

(46 reference statements)

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“…We remark that a similar approach was used by the authors in [24] to answer a question of Sokal concerning the location of zeros of the independence polynomial, a.k.a., the partition function of the hard-core model.…”

Section: Our Main Results Arementioning

confidence: 95%

Location of zeros for the partition function of the Ising model on bounded degree graphs

Peters

Regts

2019

J. London Math. Soc.

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The seminal Lee-Yang theorem states that for any graph the zeros of the partition function of the ferromagnetic Ising model lie on the unit circle in C. In fact the union of the zeros of all graphs is dense on the unit circle. In this paper we study the location of the zeros for the class of graphs of bounded maximum degree d ≥ 3, both in the ferromagnetic and the anti-ferromagnetic case. We determine the location exactly as a function of the inverse temperature and the degree d. An important step in our approach is to translate to the setting of complex dynamics and analyze a dynamical system that is naturally associated to the partition function.MSC2010: 37F10, 05C31, 68W25, 82B20. † Supported by a personal NWO Veni grant.

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Section: Our Main Results Arementioning

confidence: 95%

Location of zeros for the partition function of the Ising model on bounded degree graphs

Peters

Regts

2019

J. London Math. Soc.

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“…It is, however, asymptotically sharp, since (∆−1) ∆−1 ∆ ∆ ∼ 1 e∆ as ∆ → ∞. For more more on zero-free regions of the hard-core partition function see [56,51].…”

Section: Convergence Of the Cluster Expansionmentioning

confidence: 99%

Algorithmic Pirogov-Sinai theory

Helmuth

Perkins

Regts

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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We develop an efficient algorithmic approach for approximate counting and sampling in the low-temperature regime of a broad class of statistical physics models on finite subsets of the lattice Z d and on the torus (Z/nZ) d . Our approach is based on combining contour representations from Pirogov-Sinai theory with Barvinok's approach to approximate counting using truncated Taylor series. Some consequences of our main results include an FPTAS for approximating the partition function of the hard-core model at sufficiently high fugacity on subsets of Z d with appropriate boundary conditions and an efficient sampling algorithm for the ferromagnetic Potts model on the discrete torus (Z/nZ) d at sufficiently low temperature. 1 n log Z T d n (λ) of the hard-core model on Z d . That is, we would like an algorithm which for any ǫ > 0 outputs a number η ∈ [f d (λ) − ǫ, f d (λ) + ǫ] and whose running time grows as slowly as possible as a function of 1/ǫ. Gamarnik and Katz [28] gave such an algorithm running in time polynomial in 1/ǫ for λ small enough that the strong spatial mixing holds; this condition implies the hard-core model is in the uniqueness regime. Adams, Briceño, Marcus, and Pavlov [1] gave a polynomial-time algorithm for approximating the free energy of the hard-core (and several other) models on Z 2 in a subset of the uniqueness regime. More interestingly, their results also apply for the hard-core and Widom-Rowlinson models on Z 2 in a subset of the non-uniqueness regime. The latter result is of a similar spirit to the results of this paper.

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“…In this paper, we continue the line of research started in [Ba16a] and continued, in particular, in [Ba17], [PR17a], [Ba16b], [PR17b], [L+17], [BR17] and [EM17], on constructing efficient algorithms for computing (approximating) combinatorially defined quantities (partition functions) by exploiting the information on their complex zeros. A typical application of the method consists of a) proving that the function in question does not have zeros in some interesting domain in C n and b) constructing a low-degree polynomial approximation for the logarithm of the function in a slightly smaller domain.…”

Section: Introduction and Main Resultsmentioning

confidence: 93%

“…Usually, part a) is where the main work is done: since there is no general method to establish that a multivariate polynomial (typically having many monomials) is non-zero in a domain in C n , some quite clever arguments are being sought and found, cf. [PR17b], [EM17], see also Section 2.5 of [Ba16b] for the very few general results in this respect. Once part a) is accomplished, part b) produces a quasipolynomial approximation algorithm in a quite straightforward way, see Section 2.2 of [Ba16b].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Computing permanents of complex diagonally dominant matrices and tensors

Barvinok

2019

Isr. J. Math.

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We prove that for any λ > 1, fixed in advance, the permanent of an n × n complex matrix, where the absolute value of each diagonal entry is at least λ times bigger than the sum of the absolute values of all other entries in the same row, can be approximated within any relative error 0 < ǫ < 1 in quasi-polynomial n O(ln n−ln ǫ) time. We extend this result to multidimensional permanents of tensors and apply it to weighted counting of perfect matchings in hypergraphs.1991 Mathematics Subject Classification. 15A15, 05C65, 41A10, 68W25, 68R05.

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On a Conjecture of Sokal Concerning Roots of the Independence Polynomial

Cited by 77 publications

References 25 publications

Location of zeros for the partition function of the Ising model on bounded degree graphs

Location of zeros for the partition function of the Ising model on bounded degree graphs

Algorithmic Pirogov-Sinai theory

Computing permanents of complex diagonally dominant matrices and tensors

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