What sets the splashback radius of dark matter haloes: accretion history or other properties?

The density profiles of dark matter haloes contain rich information about their growth history and physical properties. One particularly interesting region is the splashback radius, $R_{\rm sp}$, which marks the transition between particles orbiting in the halo and particles undergoing first infall. While the dependence of $R_{\rm sp}$ on the recent accretion rate is well established and theoretically expected, it is not clear exactly what parts of the accretion history $R_{\rm sp}$ responds to, and what other halo properties might additionally influence its position. We comprehensively investigate these questions by correlating the dynamically measured splashback radii of a large set of simulated haloes with their individual growth histories as well as their structural, dynamical, and environmental properties. We find that $R_{\rm sp}$ is sensitive to the accretion over one crossing time but largely insensitive to the prior history (in contrast to concentration, which probes earlier epochs). All secondary correlations are much weaker, but we discern a relatively higher $R_{\rm sp}$ in less massive, older, more elliptical, and more tidally deformed haloes. Despite these minor influences, we conclude that the splashback radius is a clean indicator of a halo's growth over the past dynamical time. We predict that the magnitude gap should be a promising observable indicator of a halo's accretion rate and splashback radius.