We present ten young ($lesssim$10 Myr) late-K and M dwarf stars observed in K2 Campaign 2 that host protoplanetary disks and exhibit quasi-periodic or aperiodic dimming events. Their optical light curves show $sim$10-20 dips in flux over the 80-day observing campaign with durations of $sim$0.5-2 days and depths of up to $sim$40%. These stars are all members of the $rho$ Ophiuchus ($sim$1 Myr) or Upper Scorpius ($sim$10 Myr) star-forming regions. To investigate the nature of these dippers we obtained: optical and near-infrared spectra to determine stellar properties and identify accretion signatures; adaptive optics imaging to search for close companions that could cause optical variations and/or influence disk evolution; and millimeter-wavelength observations to constrain disk dust and gas masses. The spectra reveal Li I absorption and H$alpha$ emission consistent with stellar youth (<50 Myr), but also accretion rates spanning those of classical and weak-line T Tauri stars. Infrared excesses are consistent with protoplanetary disks extending to within $sim$10 stellar radii in most cases; however, the sub-mm observations imply disk masses that are an order of magnitude below those of typical protoplanetary disks. We find a positive correlation between dip depth and WISE-2 excess, which we interpret as evidence that the dipper phenomenon is related to occulting structures in the inner disk, although this is difficult to reconcile with the weakly accreting aperiodic dippers. We consider three mechanisms to explain the dipper phenomenon: inner disk warps near the co-rotation radius related to accretion; vortices at the inner disk edge produced by the Rossby Wave Instability; and clumps of circumstellar material related to planetesimal formation.
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