The Limit of the Zero Locus of the Independence Polynomial for Bounded Degree Graphs

The goal of this paper is to accurately describe the maximal zero-free region of the independence polynomial for graphs of bounded degree, for large degree bounds. In the previous work with de Boer, Guerini, and Regts, it was demonstrated that this zero-free region coincides with the normality region of the related occupation ratios. These ratios form a discrete semigroup that is in a certain sense generated by finitely many rational maps. We will show that as the degree bound converges to infinity, the properly rescaled normality regions converge to a limit domain, which can be described as the maximal boundedness component of a semigroup generated by infinitely many exponential maps.
We prove that away from the real axis this boundedness component avoids a neighborhood of the boundary of the limit cardioid, answering a recent question by Andreas Galanis. We also give an exact formula for the boundary of the boundedness component near the positive real boundary point.