2015 IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS) (2015)

Berkeley, CA, USA

Oct. 17, 2015 to Oct. 20, 2015

ISSN: 0272-5428

ISBN: 978-1-4673-8191-8

pp: 1231-1245

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/FOCS.2015.79

ABSTRACT

We study the computational complexity of several natural problems arising in statistical physics and combinatorics. In particular, we consider the following problems: the mean magnetization and mean energy of the Ising model (both the ferromagnetic and the anti-ferromagnetic settings), the average size of an independent set in the hard core model, and the average size of a matching in the monomer-dimer model. We prove that for all non-trivial values of the underlying model parameters, exactly computing these averages is #P-hard. In contrast to previous results of Sinclair and Srivastava (2013) for the mean magnetization of the ferromagnetic Ising model, our approach does not use any Lee-Yang type theorems about the complex zeros of partition functions. Indeed, it was due to the lack of suitable Lee-Yang theorems for models such as the anti-ferromagnetic Ising model that some of the problems we study here were left open by Sinclair and Srivastava. In this paper, we instead use some relatively simple and well-known ideas from the theory of automatic symbolic integration to complete our hardness reductions.

INDEX TERMS

Computational modeling, Magnetization, Physics, Computational complexity, Magnetic cores, Interpolation

CITATION

Leonard J. Schulman,
Alistair Sinclair,
Piyush Srivastava,
"Symbolic Integration and the Complexity of Computing Averages",

*2015 IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS)*, vol. 00, no. , pp. 1231-1245, 2015, doi:10.1109/FOCS.2015.79