Abstract
We define the chromatic measure of a finite simple graph as the uniform distribution on its chromatic roots. We show that for a Benjamini-Schramm convergent sequence of finite graphs, the chromatic measures converge in holomorphic moments.
As a corollary, for a convergent sequence of finite graphs, we prove that the normalized log of the chromatic polynomial converges to an analytic function outside a bounded disc. This generalizes a recent result of Borgs, Chayes, Kahn and Lovász, who proved convergence at large enough positive integers and answers a question of Borgs.
Our methods also lead to explicit estimates on the number of proper colorings of graphs with large girth.
Similar content being viewed by others
References
A. Bandyopadhyay and D. Gamarnik: Counting without sampling. Asymptotics of the logpartition function for certain statistical physics models, Random Structures & Algorithms 33 (2008), 452–479.
C. Borgs: Absence of Zeros for the Chromatic Polynomial on Bounded Degree Graphs, Combinatorics, Probability and Computing 15 (2006), 63–74.
I. Benjamini and O. Schramm: Recurrence of distributional limits of finite planar graphs, Electron. J. Probab. 6 (2001), 1–13.
C. Borgs, J. Chayes, J. Kahn and L. Lovász: Left and right convergence of graphs with bounded degree, Random Structures & Algorithms 42 (2013).
P. Csikvári and P. E. Frenkel: Benjamini-Schramm continuity of root moments of graph polynomials, http://arxiv.org/abs/1204.0463
L. Lovász: Large Networks and Graph Limits. Colloquium Publications, vol. 60. American Mathematical Society (2012)
R. Lyons: Asymptotic enumeration of spanning trees, Combinatorics, Probability and Computing 14 (2005), 491–522.
S. N. Mergelyan: Uniform approximations to functions of a complex variable, Uspehi Mat. Nauk (N.S.) 7 (1952), 31–122.
A. Procacci, B. Scoppola and V. Gerasimov: Potts model on infinite graphs and the limit of chromatic polynomials, Commun. Math. Phys. 235 (2003), 215–231.
G. C. Rota: On the foundations of combinatorial theory I. Theory of Möbius functions, Probability theory and related flelds 2 (1964), 340–368.
J. Salas and A. D. Sokal: Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models V. Further Results for the Square-Lattice Chromatic Polynomial, J. Stat. Phys. 135 (2009), 279–373.
A. D. Sokal: Bounds on the complex zeros of (di)chromatic polynomials and Potts-model partition functions, Combinatorics, Probability and Computing 10 (2001), 41–77.
A. D. Sokal: The multivariate Tutte polynomial (alias Potts model) for graphs and matroids, in: Surveys in Combinatorics (Webb, BS, ed.), 2005, 173–226. Cambridge University Press.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Abért, M., Hubai, T. Benjamini-Schramm convergence and the distribution of chromatic roots for sparse graphs. Combinatorica 35, 127–151 (2015). https://doi.org/10.1007/s00493-014-3066-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00493-014-3066-7