Asymptotics of quantum spin networks at a fixed root of unity

Details Asymptotics of quantum spin networks at a fixed root of unity Asymptotics of quantum spin networks at a fixed root of unity

2010-03-25

arxiv.org/abs/1003.4954v3

Mathematics

Geometric Topology

19 pages, 14 figures

Scientific paper

A classical spin network consists of a ribbon graph (i.e., an abstract graph with a cyclic ordering of the vertices around each edge) and an admissible coloring of its edges by natural numbers. The standard evaluation of a spin network is an integer number. In a previous paper, we proved an existence theorem for the asymptotics of the standard evaluation of an arbitrary classical spin network when the coloring of its edges are scaled by a large natural number. In the present paper, we extend the results to the case of an evaluation of quantum spin networks of arbitrary valency at a fixed root of unity. As in the classical case, our proofs use the theory of $G$-functions of Andr\'e, together with some new results concerning holonomic and $q$-holonomic sequences of Wilf-Zeilberger.

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