Quantum invariants of random 3-manifolds

TL;DR

This paper studies the statistical behavior of quantum invariants of random 3-manifolds, revealing a universal Rayleigh distribution for their absolute values and implications for topological properties.

Contribution

It demonstrates that the absolute value of SO(3) Witten-Reshetikhin-Turaev invariants follows a Rayleigh distribution for prime levels, regardless of the manifold's Heegaard genus.

Findings

Quantum invariants follow a Rayleigh distribution for prime levels.

Probability of invariants certifying Heegaard genus is very small.

Distribution is consistent for surface bundles over the circle.

Abstract

We consider the SO(3) Witten-Reshetikhin-Turaev quantum invariants of random 3-manifolds. When the level r is prime, we show that the asymptotic distribution of the absolute value of these invariants is given by the standard Rayleigh distribution and independent of the choice of level. Hence the probability that the quantum invariant certifies the Heegaard genus of a random 3-manifold of a fixed Heegaard genus g is positive but very small, less than 10^-7 except when g < 4. We also examine random surface bundles over the circle and find the same distribution for quantum invariants there.