Asymptotics of quantum spin networks at a fixed root of unity

TL;DR

This paper extends the asymptotic analysis of quantum spin networks evaluated at a fixed root of unity, using advanced mathematical tools to understand their behavior as edge colorings grow large.

Contribution

It introduces new asymptotic results for quantum spin networks at roots of unity, expanding previous classical results to the quantum case with arbitrary valency.

Findings

Established asymptotic formulas for quantum spin networks at roots of unity

Extended classical spin network asymptotics to quantum case

Utilized G-functions and holonomic sequence theory in proofs

Abstract

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.