  We provide upper and lower bounds on the escape rate of the Bhattacharyya
process corresponding to polar codes and transmission over the the binary
erasure channel. More precisely, we bound the exponent of the number of
sub-channels whose Bhattacharyya constant falls in a fixed interval $[a,b]$.
Mathematically this can be stated as bounding the limit $\lim_{n \to \infty}
\frac{1}{n} \ln \mathbb{P}(Z_n \in [a,b])$, where $Z_n$ is the Bhattacharyya
process. The quantity $\mathbb{P}(Z_n \in [a,b])$ represents the fraction of
sub-channels that are still un-polarized at time $n$.
