The hyperbolicity and linear expected-volume conjecture for random Heegaard splittings

Let $N_\phi$ be the 3-manifold obtained from a random Heegaard splitting determined by a mapping class $\phi$, and let $L$ denote the random-walk length. The hyperbolicity and linear expected-volume conjecture. As $L\to\infty$, the probability that $N_\phi$ is hyperbolic tends to $1$, and the expected volume of $N_\phi$ grows linearly in $L$. This predicts that long random Heegaard splittings are typically hyperbolic and have volumes of the scale suggested by their random-walk length. The source gives no resolution status.