Asymptotics of classical spin networks

Abstract

A spin network is a cubic ribbon graph labeled by representations of $SU\left(2\right)$. Spin networks are important in various areas of Mathematics ($3$–dimensional Quantum Topology), Physics (Angular Momentum, Classical and Quantum Gravity) and Chemistry (Atomic Spectroscopy). The evaluation of a spin network is an integer number. The main results of our paper are: (a) an existence theorem for the asymptotics of evaluations of arbitrary spin networks (using the theory of $G$–functions), (b) a rationality property of the generating series of all evaluations with a fixed underlying graph (using the combinatorics of the chromatic evaluation of a spin network), (c) rigorous effective computations of our results for some $6j$–symbols using the Wilf–Zeilberger theory and (d) a complete analysis of the regular Cube $12j$ spin network (including a nonrigorous guess of its Stokes constants), in the appendix.