On the Correlation Between Polarized BECs

We consider the $2^n$ 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.