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Lee–Yang Theorems and the Complexity of Computing Averages

Abstract

We study the complexity of computing average quantities related to spin systems, such as the mean magnetization and susceptibility in the ferromagnetic Ising model, and the average dimer count (or average size of a matching) in the monomer-dimer model. By establishing connections between the complexity of computing these averages and the location of the complex zeros of the partition function, we show that these averages are #P-hard to compute, and hence, under standard assumptions, computationally intractable. In the case of the Ising model, our approach requires us to prove an extension of the famous Lee–Yang Theorem from the 1950s.

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Correspondence to Piyush Srivastava.

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Communicated by F. Toninelli

An extended abstract of this paper appeared in the proceedings of the annual ACM Symposium on the Theory of Computing (STOC), 2013, pp. 625–634.

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Sinclair, A., Srivastava, P. Lee–Yang Theorems and the Complexity of Computing Averages. Commun. Math. Phys. 329, 827–858 (2014). https://doi.org/10.1007/s00220-014-2036-7

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  • DOI: https://doi.org/10.1007/s00220-014-2036-7

Keywords

  • Partition Function
  • Connected Graph
  • Ising Model
  • Polynomial Time Algorithm
  • Hamiltonian Path