Fisher zeros and correlation decay in the Ising model

TL;DR

This paper investigates Fisher zeros of the Ising model's partition function, showing they do not exist near the correlation decay region, linking phase transition notions and aiding efficient approximation methods.

Contribution

It establishes the absence of Fisher zeros in the correlation decay region for the zero-field Ising model, connecting complex zeros with correlation decay and computational approximation.

Findings

Fisher zeros are absent near the correlation decay region.

The result links phase transition concepts with analyticity of the free energy.

Implications for efficient deterministic partition function approximation.

Abstract

We study the complex zeros of the partition function of the Ising model, viewed as a polynomial in the "interaction parameter"; these are known as Fisher zeros in light of their introduction by Fisher in 1965. While the zeros of the partition function as a polynomial in the "field" parameter have been extensively studied since the classical work of Lee and Yang, comparatively little is known about Fisher zeros for general graphs. Our main result shows that the zero-field Ising model has no Fisher zeros in a complex neighborhood of the entire region of parameters where the model exhibits correlation decay. In addition to shedding light on Fisher zeros themselves, this result also establishes a formal connection between two distinct notions of phase transition for the Ising model: the absence of complex zeros (analyticity of the free energy, or the logarithm of the partition function) and decay of correlations with distance. We also discuss the consequences of our result for efficient deterministic approximation of the partition function. Our proof relies heavily on algorithmic techniques, notably Weitz's self-avoiding walk tree, and as such belongs to a growing body of work that uses algorithmic methods to resolve classical questions in statistical physics.