We consider the Witten–Reshetikhin–Turaev quantum invariants of random 3–manifolds. When the level 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 is positive but very small, less than except when . We also examine random surface bundles over the circle and find the same distribution for quantum invariants there.
Nathan M Dunfield. Helen Wong. "Quantum invariants of random 3–manifolds." Algebr. Geom. Topol. 11 (4) 2191 - 2205, 2011. https://doi.org/10.2140/agt.2011.11.2191