  Consider the transmission of a polar code of block length $N$ and rate $R$
over a binary memoryless symmetric channel $W$ and let $P_e$ be the block error
probability under successive cancellation decoding. In this paper, we develop
new bounds that characterize the relationship of the parameters $R$, $N$,
$P_e$, and the quality of the channel $W$ quantified by its capacity $I(W)$ and
its Bhattacharyya parameter $Z(W)$.
  In previous work, two main regimes were studied. In the error exponent
regime, the channel $W$ and the rate $R<I(W)$ are fixed, and it was proved that
the error probability $P_e$ scales roughly as $2^{-\sqrt{N}}$. In the scaling
exponent approach, the channel $W$ and the error probability $P_e$ are fixed
and it was proved that the gap to capacity $I(W)-R$ scales as $N^{-1/\mu}$.
Here, $\mu$ is called scaling exponent and this scaling exponent depends on the
channel $W$. A heuristic computation for the binary erasure channel (BEC) gives
$\mu=3.627$ and it was shown that, for any channel $W$, $3.579 \le \mu \le
5.702$.
  Our contributions are as follows. First, we provide the tighter upper bound
$\mu \le 4.714$ valid for any $W$. With the same technique, we obtain $\mu \le
3.639$ for the case of the BEC, which approaches very closely its heuristically
derived value. Second, we develop a trade-off between the gap to capacity
$I(W)-R$ and the error probability $P_e$ as functions of the block length $N$.
In other words, we consider a moderate deviations regime in which we study how
fast both quantities, as functions of the block length $N$, simultaneously go
to $0$. Third, we prove that polar codes are not affected by error floors. To
do so, we fix a polar code of block length $N$ and rate $R$. Then, we vary the
channel $W$ and we show that the error probability $P_e$ scales as the
Bhattacharyya parameter $Z(W)$ raised to a power that scales roughly like
$\sqrt{N}$.
