Circumstellar disks are thought to experience a rapid "transition" phase in their evolution that can have a considerable impact on the formation and early development of planetary systems. We present new and archival high angular resolution (0farcs3 ? 40-75 AU) Submillimeter Array (SMA) observations of the 880 ?m (340 GHz) dust continuum emission from 12 such transition disks in nearby star-forming regions. In each case, we directly resolve a dust-depleted disk cavity around the central star. Using two-dimensional Monte Carlo radiative transfer calculations, we interpret these dust disk structures in a homogeneous, parametric model framework by reproducing their SMA continuum visibilities and spectral energy distributions. The cavities in these disks are large (R cav = 15-73 AU) and substantially depleted of small (~?m-sized) dust grains, although their mass contents are still uncertain. The structures of the remnant material at larger radii are comparable to normal disks. We demonstrate that these large cavities are relatively common among the millimeter-bright disk population, comprising at least 1 in 5 (20%) of the disks in the bright half (and >=26% of the upper quartile) of the millimeter luminosity (disk mass) distribution. Utilizing these results, we assess some of the physical mechanisms proposed to account for transition disk structures. As has been shown before, photoevaporation models do not produce the large cavity sizes, accretion rates, and disk masses representative of this sample. A sufficient decrease of the dust optical depths in these cavities by particle growth would be difficult to achieve: substantial growth (to meter sizes or beyond) must occur in large (tens of AU) regions of low turbulence without also producing an abundance of small particles. Given those challenges, we suggest instead that the observations are most commensurate with dynamical clearing due to tidal interactions with low-mass companions—very young (~1 Myr) brown dwarfs or giant planets on long-period orbits.
