Benjamini-Schramm convergence and the distribution of chromatic roots for sparse graphs
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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.
Mathematics Subject Classification (2000)05C31 05C60 05C15 82B20
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