  A non-zero mutual information between morphology of a galaxy and its
large-scale environment is known to exist in SDSS upto a few tens of Mpc. It is
important to test the statistical significance of these mutual information if
any. We propose three different methods to test the statistical significance of
these non-zero mutual information and apply them to SDSS and Millennium Run
simulation. We randomize the morphological information of SDSS galaxies without
affecting their spatial distribution and compare the mutual information in the
original and randomized datasets. We also divide the galaxy distribution into
smaller subcubes and randomly shuffle them many times keeping the morphological
information of galaxies intact. We compare the mutual information in the
original SDSS data and its shuffled realizations for different shuffling
lengths. Using a t-test, we find that a small but statistically significant (at
99.9% confidence level) mutual information between morphology and environment
exists upto the entire length scale probed. We also conduct another experiment
using mock datasets from a semi-analytic galaxy catalogue where we assign
morphology to galaxies in a controlled manner based on the density at their
locations. The experiment clearly demonstrate that mutual information can
effectively capture the physical correlations between morphology and
environment. Our analysis suggests that physical association between morphology
and environment may extend to much larger length scales than currently believed
and the information theoretic framework presented here, can serve as a
sensitive and useful probe of the assembly bias and large-scale environmental
dependence of galaxy properties.
