  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.
