Asymptotics of classical spin networks

TL;DR

This paper investigates the asymptotic behavior of classical spin networks, providing existence theorems, rationality properties, computational methods, and detailed analysis of specific networks, bridging mathematics, physics, and chemistry.

Contribution

It introduces new theoretical results on the asymptotics of spin network evaluations, including existence theorems, rationality of generating series, and computational techniques.

Findings

Existence theorem for asymptotics of spin networks

Rationality of generating series for fixed graphs

Effective computations for 6j-symbols and analysis of the Cube 12j network

Abstract

A spin network is a cubic ribbon graph labeled by representations of $\mathrm{SU}(2)$. 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 non-rigorous guess of its Stokes constants), in the appendix.