Associated to any finite simple graph is the chromatic polynomial whose complex zeros are called the chromatic zeros of . A hierarchical lattice is a sequence of finite simple graphs built recursively using a substitution rule expressed in terms of a generating graph. For each , let denote the probability measure that assigns a Dirac measure to each chromatic zero of . Under a mild hypothesis on the generating graph, we prove that the sequence converges to some measure as tends to infinity. We call the limiting measure of chromatic zeros associated to . In the case of the diamond hierarchical lattice we prove that the support of has Hausdorff dimension two.
The main techniques used come from holomorphic dynamics and more specifically the theories of activity/bifurcation currents and arithmetic dynamics. We prove a new equidistribution theorem that can be used to relate the chromatic zeros of a hierarchical lattice to the activity current of a particular marked point. We expect that this equidistribution theorem will have several other applications.
Cite this article
Ivan Chio, Roland K.W. Roeder, Chromatic zeros on hierarchical lattices and equidistribution on parameter space. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 8 (2021), no. 4, pp. 491–536DOI 10.4171/AIHPD/109