2025/11/13 by Chang, Shu-Chiuan, Shrock, Robert
#FOS: Physical sciences #Statistical Mechanics (cond-mat.stat-mech)
paper · doi:10.48550/arxiv.2511.10405
We calculate the continuous accumulation set \cal Bq(p,ℓ) of zeros of the chromatic polynomial P(G(p,ℓ)m,q) in the limit m → ∞, on a family of graphs G(p,ℓ)m defined such that G(p,ℓ)m is obtained from G(p,ℓ)m-1 by replacing each edge (i.e., bond) on G(p,ℓ)m by p paths each of length ℓ edges, starting with the tree graph T2. Our method uses the property that the chromatic polynomial P(G,q) of a graph G is equal to the v=-1 evaluation of the partition function of the q-state Potts model, together with (i) the property that Z(G(p,ℓ)m,q,v) can be expressed via an exact closed-form real-space renormalization (RG) group transformation in terms of Z(G(p,ℓ)m-1,q,v'), where v'=F(p,ℓ),q(v) is a rational function of v and q and (ii) \cal Bq(p,ℓ)(v) is the locus in the complex q-plane that separates regions of different asymptotic behavior of the m-fold iterated RG transformation F(p,ℓ),q(v) in the m → ∞ limit. Thus, our results involve calculations of region diagrams in the complex q-plane showing the type of behavior that occurs in the m → ∞ limit of the m-fold iterated RG transformation mapping F(p,ℓ),q(v) starting with the initial value v=v0=-1. Calculations are presented of the maximal point qc(G(p,ℓ)_∞) at which the locus \cal Bq crosses the real-q axis, as well as several other points at which, depending on p and ℓ, the locus \cal Bq crosses this axis. We give explicit results for a variety of (p,ℓ) cases and observe a number of interesting features. Calculations of the ground-state degeneracy of the Potts antiferromagnet at qc(G(p,ℓ)_∞) are presented. This work extends a previous study with R. Roeder of the (p,ℓ)=(2,2) case to higher p and ℓ values.