  For a binary-input memoryless symmetric channel $W$, we consider the
asymptotic behavior of the polarization process in the large block-length
regime when transmission takes place over $W$. In particular, we study the
asymptotics of the cumulative distribution $\mathbb{P}(Z_n \leq z)$, where
$\{Z_n\}$ is the Bhattacharyya process defined from $W$, and its dependence on
the rate of transmission. On the basis of this result, we characterize the
asymptotic behavior, as well as its dependence on the rate, of the block error
probability of polar codes using the successive cancellation decoder. This
refines the original bounds by Ar{\i}kan and Telatar. Our results apply to
general polar codes based on $\ell \times \ell$ kernel matrices.
  We also provide lower bounds on the block error probability of polar codes
using the MAP decoder. The MAP lower bound and the successive cancellation
upper bound coincide when $\ell=2$, but there is a gap for $\ell>2$.
