We consider the 2n channels synthesized by the n-fold application of
Ar\i{}kan's polar transform to a binary erasure channel (BEC). The synthetic
channels are BECs themselves, and we show that, asymptotically for almost all
these channels, the pairwise correlations between their erasure events are
extremely small: the correlation coefficients vanish faster than any
exponential in n. Such a fast decay of correlations allows us to conclude
that the union bound on the block error probability of polar codes is very
tight.Comment: 9 pages, Extended version of a paper submitted to ISIT 201