  We show that the entire class of polar codes (up to a natural necessary
condition) converge to capacity at block lengths polynomial in the gap to
capacity, while simultaneously achieving failure probabilities that are
exponentially small in the block length (i.e., decoding fails with probability
$\exp(-N^{\Omega(1)})$ for codes of length $N$). Previously this combination
was known only for one specific family within the class of polar codes, whereas
we establish this whenever the polar code exhibits a condition necessary for
any polarization. Our results adapt and strengthen a local analysis of polar
codes due to the authors with Nakkiran and Rudra [Proc. STOC 2018]. Their
analysis related the time-local behavior of a martingale to its global
convergence, and this allowed them to prove that the broad class of polar codes
converge to capacity at polynomial block lengths. Their analysis easily adapts
to show exponentially small failure probabilities, provided the associated
martingale, the ``Arikan martingale'', exhibits a corresponding strong local
effect. The main contribution of this work is a much stronger local analysis of
the Arikan martingale. This leads to the general result claimed above. In
addition to our general result, we also show, for the first time, polar codes
that achieve failure probability $\exp(-N^{\beta})$ for any $\beta < 1$ while
converging to capacity at block length polynomial in the gap to capacity.
Finally we also show that the ``local'' approach can be combined with any
analysis of failure probability of an arbitrary polar code to get essentially
the same failure probability while achieving block length polynomial in the gap
to capacity.
