  We study a resource allocation setting where $m$ discrete items are to be
divided among $n$ agents with additive utilities, and the agents' utilities for
individual items are drawn at random from a probability distribution. Since
common fairness notions like envy-freeness and proportionality cannot always be
satisfied in this setting, an important question is when allocations satisfying
these notions exist. In this paper, we close several gaps in the line of work
on asymptotic fair division. First, we prove that the classical round-robin
algorithm is likely to produce an envy-free allocation provided that
$m=\Omega(n\log n/\log\log n)$, matching the lower bound from prior work. We
then show that a proportional allocation exists with high probability as long
as $m\geq n$, while an allocation satisfying envy-freeness up to any item (EFX)
is likely to be present for any relation between $m$ and $n$. Finally, we
consider a related setting where each agent is assigned exactly one item and
the remaining items are left unassigned, and show that the transition from
non-existence to existence with respect to envy-free assignments occurs at
$m=en$.
