  Motivated by the Beck-Fiala conjecture, we study discrepancy bounds for
random sparse set systems. Concretely, these are set systems $(X,\Sigma)$,
where each element $x \in X$ lies in $t$ randomly selected sets of $\Sigma$,
where $t$ is an integer parameter. We provide new bounds in two regimes of
parameters. We show that when $|\Sigma| \ge |X|$ the hereditary discrepancy of
$(X,\Sigma)$ is with high probability $O(\sqrt{t \log t})$; and when $|X| \gg
|\Sigma|^t$ the hereditary discrepancy of $(X,\Sigma)$ is with high probability
$O(1)$. The first bound combines the Lov{\'a}sz Local Lemma with a new argument
based on partial matchings; the second follows from an analysis of the lattice
spanned by sparse vectors.
