  We explore the relation between the stellar mass surface density and the mass
surface density of molecular hydrogen gas in twelve nearby molecular clouds
that are located at $<$1.5 kpc distance. The sample clouds span an order of
magnitude range in mass, size, and star formation rates. We use thermal dust
emission from $Herschel$ maps to probe the gas surface density and the young
stellar objects from the most recent $Spitzer$ Extended Solar Neighborhood
Archive (SESNA) catalog to probe the stellar surface density. Using a
star-sampled nearest neighbor technique to probe the star-gas surface density
correlations at the scale of a few parsecs, we find that the stellar mass
surface density varies as a power-law of the gas mass surface density, with a
power-law index of $\sim$2 in all the clouds. The consistent power-law index
implies that star formation efficiency is directly correlated with gas column
density, and no gas column density threshold for star formation is observed. We
compare the observed correlations with the predictions from an analytical model
of thermal fragmentation, and with the synthetic observations of a recent
hydrodynamic simulation of a turbulent star-forming molecular cloud. We find
that the observed correlations are consistent for some clouds with the thermal
fragmentation model and can be reproduced using the hydrodynamic simulations.
