Generating series and asymptotics of classical spin networks
This paper investigates two aspects of classical spin networks with group \(SU(2)\). First, using Gaussian integrals, the paper generalizes a formula of Westbury to the case of spin networks whose edges are labeled by holonomies. A combinatorial interpretation of the formula is also provided. Second, subject to certain conditions, a formula is given for the asymptotic behaviour of the generating series of squares of spin networks as labels are rescaled by a constant diverging to infinity. Such asymptotic behaviour (proven to exist by \textit{S. Garoufalidis} and \textit{R. van der Veen} [Geom. Topol. 17, No. 1, 1--37 (2013; Zbl 1277.57020)]) is expected to recover Euclidean geometric quantities, and this type of formula is a holy grail for the study of spin networks. However, it is not known how to describe when the conditions hold for the formula and indeed no examples have been shown in which these hypotheses hold other than for the \(6j\)-symbols.
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