Motion of Lee–Yang Zeros

We consider the zeros of partition function Ising model with ferromagnetic pair interactions and complex external field. Under assumption that graph strictly positive is connected, we vary interaction (denoted by $t$) at a fixed edge. It already known each zero monotonic (either increasing or decreasing) in $t$; prove its motion local: entire trajectories any two distinct are disjoint. If underlying complete all take same value $t\geq 0$ (i.e., Curie-Weiss model), principal (those $i[0,\pi/2)$) decrease $t$.