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,#P-hardness, Computational Complexity, Statistical Mechanics, Counting Problems

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