Local patch analysis for testing statistical isotropy of the Planck
  convergence map
  The small but measurable effect of weak gravitational lensing on the cosmic
microwave background radiation provide information about the large-scale
distribution of matter in the universe. We use the all sky distribution of
matter, as represented by the {\em convergence map} that is inferred from CMB
lensing measurement by Planck survey, to test the fundamental assumption of
Statistical Isotropy (SI) of the universe. For the analysis we use the $\alpha$
statistic that is devised from the contour Minkowski tensor, a tensorial
generalization of the scalar Minkowski functional, the contour length. In
essence, the $\alpha$ statistic captures the ellipticity of isofield contours
at any chosen threshold value of a smooth random field and provides a measure
of anisotropy. The SI of the observed convergence map is tested against the
suite of realistic simulations of the convergence map provided by the Planck
collaboration. We first carry out a global analysis using the full sky data
after applying the galactic and point sources mask. We find that the observed
data is consistent with SI. Further we carry out a local search for departure
from SI in small patches of the sky using $\alpha$. This analysis reveals
several sky patches which exhibit deviations from simulations with statistical
significance higher than 95\% confidence level (CL). Our analysis indicates
that the source of the anomalous behaviour of most of the outlier patches is
inaccurate estimation of noise. We identify two outlier patches which exhibit
anomalous behaviour originating from departure from SI at higher than 95\% CL.
Most of the anomalous patches are found to be located roughly along the
ecliptic plane or in proximity to the ecliptic poles.
