Structure and motion of the Lee-Yang zeros

For an Ising model satisfying the Lee-Yang condition, the zeros of the partition function Z and those of the associated functions Z_A in the space of imaginary magnetic fields at all lattice sites are determined by a single analytic hypersurface. The sense of motion of the zeros of Z as the interactions are varied can be related to the positions of the zeros of the Z_A. Contrary to a plausible conjecture, it is not true that all of the zeros of Z in a uniform field tend towards the point ẑ = 1 in the complex fugacity plane as the temperature is lowered, but it is possible that the first zero (that nearest to ẑ = 1) has a monotone motion. Various simplicity and intertwining properties of the zeros of Z and Z_A which generalize earlier results are proved by a new argument which makes direct use of the Lee-Yang property.