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The Ising Partition Function: Zeros and Deterministic Approximation

Jingcheng Liu, Alistair Sinclair, Piyush Srivastava
2018 Journal of statistical physics  
We study the problem of approximating the partition function of the ferromagnetic Ising model in graphs and hypergraphs.  ...  Our first result is a deterministic approximation scheme (an FPTAS) for the partition function in bounded degree graphs that is valid over the entire range of parameters β (the interaction) and λ (the  ...  Perhaps surprisingly, however, no deterministic approximation algorithm is known for the classical ferromagnetic Ising partition function that works over anything close to the full range of the randomized  ... 
doi:10.1007/s10955-018-2199-2 fatcat:c3ikrhtexbgvpa346qlklahysy

More on zeros and approximation of the Ising partition function [article]

Alexander Barvinok, Nicholas Barvinok
2021 arXiv   pre-print
We consider the problem of computing the partition function ∑_x e^f(x), where f: {-1, 1}^n ⟶ 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 < ϵ < 1 in quasi-polynomial n^O(ln n - lnϵ) time if the Lipschitz constant of the  ...  Acknowledgment The authors are grateful to Alistair Sinclair for answering questions regarding correlation decay in the Ising model.  ... 
arXiv:2005.11232v3 fatcat:vl6desixofb3fbvzhwmogmvaim

The complexity of approximating complex-valued Ising and Tutte partition functions [article]

Leslie Ann Goldberg, Heng Guo
2017 arXiv   pre-print
We study the complexity of approximately evaluating the Ising and Tutte partition functions with complex parameters.  ...  The motivation for this paper is to study more comprehensively the complexity of (classically) approximating the Ising and Tutte partition functions with complex parameters.  ...  Acknowledgements We thank Dan Shepherd and Mark Jerrum for useful discussions.  ... 
arXiv:1409.5627v4 fatcat:vzlt7m7gx5cu7k45gnabaxvjoq

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

Han Peters, Guus Regts
2019 arXiv   pre-print
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.  ...  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.  ...  We moreover thank the anonymous referees for helpful comments improving the presentation of the paper as well as for suggesting a number of important references.  ... 
arXiv:1810.01699v2 fatcat:wnx7kpkclzdu7jsemuyzz5auty

More on zeros and approximation of the Ising partition function

Alexander Barvinok, Nicholas Barvinok
2021 Forum of Mathematics, Sigma  
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  ...  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 $\{-1, 1\}^n$  ...  , as opposed to quasi-polynomial, if the degree of the underlying graph is bounded from above in advance, and to the anonymous referees for their careful reading of the paper and catching inaccuracies.  ... 
doi:10.1017/fms.2021.40 fatcat:it4d2qivtfbk3nwhdc5wyn2one

Polynomial constraint satisfaction problems, graph bisection, and the Ising partition function

Alexander D. Scott, Gregory B. Sorkin
2009 ACM Transactions on Algorithms  
Where the usual CSPs from computer science and optimization have real-valued score functions, and partition functions from physics have monomials, PCSP has scores that are arbitrary multivariate formal  ...  This gives the first polynomial-space exact algorithm more efficient than exhaustive enumeration for the well-studied problems of finding a minimum bisection of a graph, and calculating the partition function  ...  Thus the partition function for I is the sum of the partition functions for the I i .  ... 
doi:10.1145/1597036.1597049 fatcat:rm2wgsxjcrfn5lkhdeslpkmnjm

The Complexity of Approximating complex-valued Ising and Tutte partition functions

Leslie Ann Goldberg, Heng Guo
2017 Computational Complexity  
We study the complexity of approximately evaluating the Ising and Tutte partition functions with complex parameters.  ...  The motivation for this paper is to study more comprehensively the complexity of (classically) approximating the Ising and Tutte partition functions with complex parameters.  ...  BQP and the Tutte polynomial Bordewich et al. [4] raised the question "of determining whether the Tutte polynomial is greater than or equal to, or less than zero at a given point."  ... 
doi:10.1007/s00037-017-0162-2 fatcat:whewokm6sbdqhpzsn47bwpym2e

Missing mass approximations for the partition function of stimulus driven Ising models

Robert Haslinger, Demba Ba, Ralf Galuske, Ziv Williams, Gordon Pipa
2013 Frontiers in Computational Neuroscience  
the stimulus driven partition function more accurately than either Monte Carlo methods or deterministic approximations.  ...  Here we present an extremely fast, yet simply implemented, method for approximating the stimulus dependent partition function in minutes or seconds.  ...  This work was supported by NIH grant K25 NS052422-02 (Robert Haslinger), the Max Planck-Gesellschaft (Ralf Galuske), and NIH grant 5R01-HD059852, PECASE and the Whitehall Foundation (Ziv Williams).  ... 
doi:10.3389/fncom.2013.00096 pmid:23898262 pmcid:PMC3721091 fatcat:mjrspxqlcnejrdc2zatfh5clte

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

Han Peters, Guus Regts
2019 Journal of the London Mathematical Society  
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.  ...  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.  ...  We thank Sander Bet, Ferenc Bencs, Pjotr Buys, David de Boer and Eoin Hurley for useful comments on a previous version of this paper.  ... 
doi:10.1112/jlms.12286 fatcat:holxjw7x5vdcjli4a7xxnufajq

The genetic code is very close to a global optimum in a model of its origin taking into account both the partition energy of amino acids and their biosynthetic relationships [article]

Massimo Di Giulio, Franco Caldararo
2021 bioRxiv   pre-print
In other words, since the partition energy is reflective of the protein structure and therefore of the enzymatic catalysis, the latter might really have been the main selective pressure that would have  ...  This might now become less sustainable, given the very high optimization that is instead observed in favor of partition energy but not polarity.  ...  approximately zero when they are arranged randomly and independently in their relative blocks.  ... 
doi:10.1101/2021.08.01.454621 fatcat:5jd556ws5vby7bsn3ds56bquiy

Polynomial-Time Approximation of Zero-Free Partition Functions [article]

Penghui Yao, Yitong Yin, Xinyuan Zhang
2022 arXiv   pre-print
Zero-free based algorithm is a major technique for deterministic approximate counting.  ...  Consequently, when the inverse temperature is close enough to zero by a constant gap, we have polynomial-time approximation algorithm for all such partition functions.  ...  It is widely believed that for various classes of partition functions of interests, the hardness of approximation is captured by the locus of complex zeros.  ... 
arXiv:2201.12772v1 fatcat:gse36s7fsbaftnfy7araf4xji4

Computational Complexity and Partition Functions (Invited Talk)

Leslie Ann Goldberg, Michael Wagner
2019 Symposium on Theoretical Aspects of Computer Science  
This paper is an extended abstract of my STACS 2019 talk "Computational Complexity and Partition Functions".  ...  1:2 Computational Complexity and Partition Functions the thresholds −λ * and λ c .  ...  There are many interesting partition functions, such as the partition functions of the Ising model and the Potts model, that will not be discussed in the talk.  ... 
doi:10.4230/lipics.stacs.2019.1 dblp:conf/stacs/Goldberg19 fatcat:572yrwejhjdz3ahb5wa6drtssa

Deterministic Polynomial-Time Approximation Algorithms for Partition Functions and Graph Polynomials

Viresh Patel, Guus Regts
2017 SIAM journal on computing (Print)  
In particular, our approach works for the Tutte polynomial and independence polynomial, as well as partition functions of complex-valued spin and edge-coloring models.  ...  More specifically, we define a large class of graph polynomials C and show that if p∈ C and there is a disk D centered at zero in the complex plane such that p(G) does not vanish on D for all bounded degree  ...  Acknowledgements We thank Alexander Barvinok for stimulating discussions, useful remarks and for sharing the results in [6] with us. We thank Andreas Galanis for informing us about [42] .  ... 
doi:10.1137/16m1101003 fatcat:3eur4cw3rfdtvcesbo55vc6kjm

Computing the partition function for graph homomorphisms with multiplicities [article]

Alexander Barvinok, Pablo Soberón
2015 arXiv   pre-print
We consider a refinement of the partition function of graph homomorphisms and present a quasi-polynomial algorithm to compute it in a certain domain.  ...  As a corollary, we obtain quasi-polynomial algorithms for computing partition functions for independent sets, perfect matchings, Hamiltonian cycles and dense subgraphs in graphs as well as for graph colorings  ...  Acknowledgments The authors are grateful to Boris Bukh for suggesting Lemma 4.1 and to anonymous referees for careful reading of the paper, suggestions and corrections.  ... 
arXiv:1410.1842v3 fatcat:ojp7bhjoaragrctczprujffg74

Unsupervised Basis Function Adaptation for Reinforcement Learning [article]

Edward Barker, Charl Ras
2019 arXiv   pre-print
When using reinforcement learning (RL) algorithms it is common, given a large state space, to introduce some form of approximation architecture for the value function (VF).  ...  Consequently there is currently interest among researchers in the potential for allowing RL algorithms to adaptively generate (i.e. to learn) approximation architectures.  ...  Acknowledgements We would like to thank the anonymous reviewers for their valuable and insightful comments.  ... 
arXiv:1703.07940v3 fatcat:4yrzhegl45affpq3opkld4se5u
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