  We present new results from a significant extension of our previous high
angular resolution (0.3" = 40 AU) Submillimeter Array survey of the 880 um
continuum emission from dusty circumstellar disks in the ~1 Myr-old Ophiuchus
star-forming region. An expanded sample is constructed to probe disk structures
that emit significantly lower millimeter luminosities (hence dust masses), down
to the median value for T Tauri stars. Using a Monte Carlo radiative transfer
code, the millimeter visibilities and broadband spectral energy distribution
for each disk are simultaneously reproduced with a two-dimensional parametric
model for a viscous accretion disk. We find wide ranges of characteristic radii
(14-198 AU) and disk masses (0.004-0.143 M_sun), but a narrow distribution of
surface density gradients (0.4-1.1) that is consistent with a uniform value
$\gamma$ = 0.9 +/- 0.2 and independent of mass (or millimeter luminosity). In
this sample, we find a correlation between the disk luminosity/mass and
characteristic radius, such that fainter disks are both smaller and less
massive. We suggest that this relationship is an imprint of the initial
conditions inherited by the disks at their formation epoch, compare their
angular momenta with those of molecular cloud cores, and speculate on how
future observations can help constrain the distribution of viscous evolution
timescales. No other correlations between disk and star properties are found.
The inferred disk structures are briefly compared with theoretical models for
giant planet formation, although resolution limitations do not permit us to
directly comment on material inside R = 20 AU. However, there is some
compelling evidence for dust evolution in the planet formation region: 4/17
disks in the sample show resolved regions of significantly reduced optical
depths within ~20-40 AU of their central stars.
