Lee-Yang theorems and the complexity of computing averages

Authors:
Alistair Sinclair;Piyush Srivastava
Affiliations:
University of California at Berkeley, Berkeley, CA, USA;University of California at Berkeley, Berkeley, CA, USA
Venue:
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Year:
2013

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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. In case of the Ising model, our approach requires us to prove an extension of the famous Lee-Yang Theorem from the 1950s.