Benjamini–Schramm continuity of root moments of graph polynomials

Recently, M. Abért and T. Hubai studied the following problem. The chromatic measure of a finite simple graph is defined to be the uniform distribution on its chromatic roots. Abért and Hubai proved that for a Benjamini–Schramm convergent sequence of finite graphs, the chromatic measures converge in holomorphic moments. They also showed that the normalized logarithm of the chromatic polynomial converges to a harmonic real function outside a bounded disc.In this paper we generalize their work to a wide class of graph polynomials, namely, multiplicative graph polynomials of bounded exponential type. A special case of our results is that for any fixed complex number v0v0 the measures arising from the Tutte polynomial ZGn(z,v0)ZGn(z,v0) converge in holomorphic moments if the sequence (Gn)(Gn) of finite graphs is Benjamini–Schramm convergent. This answers a question of Abért and Hubai in the affirmative. Even in the original case of the chromatic polynomial, our proof is considerably simpler.