  We present a method for the computation of the variance of cosmic microwave
background (CMB) temperature maps on azimuthally symmetric patches using a fast
convolution approach. As an example of the application of the method, we show
results for the search for concentric rings with unusual variance in the 7-year
WMAP data. We re-analyse claims concerning the unusual variance profile of
rings centred at two locations on the sky that have recently drawn special
attention in the context of the conformal cyclic cosmology scenario proposed by
Penrose (2009). We extend this analysis to rings with larger radii and centred
on other points of the sky. Using the fast convolution technique enables us to
perform this search with higher resolution and a wider range of radii than in
previous studies. We show that for one of the two special points rings with
radii larger than 10 degrees have systematically lower variance in comparison
to the concordance LambdaCDM model predictions. However, we show that this
deviation is caused by the multipoles up to order l=7. Therefore, the deficit
of power for concentric rings with larger radii is yet another manifestation of
the well-known anomalous CMB distribution on large angular scales. Furthermore,
low variance rings can be easily found centred on other points in the sky. In
addition, we show also the results of a search for extremely high variance
rings. As for the low variance rings, some anomalies seem to be related to the
anomalous distribution of the low-order multipoles of the WMAP CMB maps. As
such our results are not consistent with the conformal cyclic cosmology
scenario.
