  Doppler planet searches have discovered that giant planets follow orbits with
a wide range of orbital eccentricities, revolutionizing theories of planet
formation. The discovery of hundreds of exoplanet candidates by NASA's Kepler
mission enables astronomers to characterize the eccentricity distribution of
small exoplanets. Measuring the eccentricity of individual planets is only
practical in favorable cases that are amenable to complementary techniques
(e.g., radial velocities, transit timing variations, occultation photometry).
Yet even in the absence of individual eccentricities, it is possible to study
the distribution of eccentricities based on the distribution of transit
durations (relative to the maximum transit duration for a circular orbit). We
analyze the transit duration distribution of Kepler planet candidates. We find
that for host stars with T_eff > 5100 K we cannot invert this to infer the
eccentricity distribution at this time due to uncertainties and possible
systematics in the host star densities. With this limitation in mind, we
compare the observed transit duration distribution with models to rule out
extreme distributions. If we assume a Rayleigh eccentricity distribution for
Kepler planet candidates, then we find best-fits with a mean eccentricity of
0.1-0.25 for host stars with T_eff < 5100 K. We compare the transit duration
distribution for different subsets of Kepler planet candidates and discuss
tentative trends with planetary radius and multiplicity. High-precision
spectroscopic follow-up observations for a large sample of host stars will be
required to confirm which trends are real and which are the results of
systematic errors in stellar radii. Finally, we identify planet candidates that
must be eccentric or have a significantly underestimated stellar radius.
