Fisher's zeros as the boundary of renormalization group flows in complex coupling spaces

We propose new methods to extend the renormalization group transformation to complex coupling spaces. We argue that Fisher's zeros are located at the boundary of the complex basin of attraction of infrared fixed points. We support this picture with numerical calculations at finite volume for two-dimensional O(N) models in the large-N limit and the hierarchical Ising model. We present numerical evidence that, as the volume increases, the Fisher's zeros of four-dimensional pure gauge SU(2) lattice gauge theory with a Wilson action stabilize at a distance larger than 0.15 from the real axis in the complex β=4/g2 plane. We discuss the implications for proofs of confinement and searches for nontrivial infrared fixed points in models beyond the standard model.