  Consider a binary-input memoryless output-symmetric channel $W$. Such a
channel has a capacity, call it $I(W)$, and for any $R<I(W)$ and strictly
positive constant $P_{\rm e}$ we know that we can construct a coding scheme
that allows transmission at rate $R$ with an error probability not exceeding
$P_{\rm e}$. Assume now that we let the rate $R$ tend to $I(W)$ and we ask how
we have to "scale" the blocklength $N$ in order to keep the error probability
fixed to $P_{\rm e}$. We refer to this as the "finite-length scaling" behavior.
This question was addressed by Strassen as well as Polyanskiy, Poor and Verdu,
and the result is that $N$ must grow at least as the square of the reciprocal
of $I(W)-R$.
  Polar codes are optimal in the sense that they achieve capacity. In this
paper, we are asking to what degree they are also optimal in terms of their
finite-length behavior. Our approach is based on analyzing the dynamics of the
un-polarized channels. The main results of this paper can be summarized as
follows. Consider the sum of Bhattacharyya parameters of sub-channels chosen
(by the polar coding scheme) to transmit information. If we require this sum to
be smaller than a given value $P_{\rm e}>0$, then the required block-length $N$
scales in terms of the rate $R < I(W)$ as $N \geq
\frac{\alpha}{(I(W)-R)^{\underline{\mu}}}$, where $\alpha$ is a positive
constant that depends on $P_{\rm e}$ and $I(W)$, and $\underline{\mu} = 3.579$.
Also, we show that with the same requirement on the sum of Bhattacharyya
parameters, the block-length scales in terms of the rate like $N \leq
\frac{\beta}{(I(W)-R)^{\overline{\mu}}}$, where $\beta$ is a constant that
depends on $P_{\rm e}$ and $I(W)$, and $\overline{\mu}=6$.
