On the correlation between polarized BECs

We consider the 2ⁿ channels synthesized by the n-fold application of Arıkan´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.