Lee-Yang theorems and the complexity of computing averages

TL;DR

This paper investigates the computational difficulty of calculating average properties in spin systems and dimer models, linking it to the zeros of the partition function and extending the Lee-Yang Theorem.

Contribution

It establishes that computing these averages is #P-hard and extends the Lee-Yang Theorem to support this complexity result.

Findings

Computing average magnetization and susceptibility is #P-hard.

Average dimer count computation is #P-hard.

Extended Lee-Yang Theorem supports complexity proofs.

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.