Quantum invariants of random 3–manifolds

Abstract

We consider the $SO\left(3\right)$ 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 a Rayleigh distribution which is 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 $g\le 3$. We also examine random surface bundles over the circle and find the same distribution for quantum invariants there.