Abstract
We study the complexity of approximating the partition function ofthe q-state Potts model and the closely related Tutte polynomialfor complex values of the underlying parameters. Apart from theclassical connections with quantum computing and phase transitionsin statistical physics, recent work in approximate counting hasshown that the behaviour in the complex plane, and more preciselythe location of zeros, is strongly connected with the complexity ofthe approximation problem, even for positive real-valued parameters.Previous work in the complex plane by Goldberg and Guo focused onq = 2, which corresponds to the case of the Ising model; for q > 2,the behaviour in the complex plane is not as well understood andmost work applies only to the real-valued Tutte plane. Our mainresult is a complete classification of the complexity of theapproximation problems for all non-real values of the parameters, byestablishing #P-hardness results that apply even when restricted toplanar graphs. Our techniques apply to all q geq 2 and furthercomplement/refine previous results both for the Ising model and theTutte plane, answering in particular a question raised by Bordewich,Freedman, Lovász and Welsh in the context of quantumcomputations.
Highlights
The q-state Potts model is a classical model of ferromagnetism in statistical physics (Potts 1952; Welsh 1993) which generalises the well-known Ising model
We study the complexity of approximating the partition function of the Potts model and the Tutte polynomial on planar graphs as the parameter y ranges in the complex plane
For the problem of exactly computing the partition function of the Potts model, Jaeger et al (1990), as a corollary of a more general classification theorem for the Tutte polynomial, established #P-hardness unless (q, y) is one of seven exceptional points, see Section 6.3 for more details; Vertigan (2005) further showed that the same classification applies on planar graphs with the exception of the Ising model (q = 2), where the problem is in FP
Summary
The q-state Potts model is a classical model of ferromagnetism in statistical physics (Potts 1952; Welsh 1993) which generalises the well-known Ising model. We study the complexity of approximating the partition function of the Potts model and the Tutte polynomial on planar graphs as the parameter y ranges in the complex plane. For the problem of exactly computing the partition function of the Potts model, Jaeger et al (1990), as a corollary of a more general classification theorem for the Tutte polynomial, established #P-hardness unless (q, y) is one of seven exceptional points, see Section 6.3 for more details; Vertigan (2005) further showed that the same classification applies on planar graphs with the exception of the Ising model (q = 2), where the problem is in FP.
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