  We initiate the probabilistic analysis of linear programming (LP) decoding of
low-density parity-check (LDPC) codes. Specifically, we show that for a random
LDPC code ensemble, the linear programming decoder of Feldman et al. succeeds
in correcting a constant fraction of errors with high probability. The fraction
of correctable errors guaranteed by our analysis surpasses previous
non-asymptotic results for LDPC codes, and in particular exceeds the best
previous finite-length result on LP decoding by a factor greater than ten. This
improvement stems in part from our analysis of probabilistic bit-flipping
channels, as opposed to adversarial channels. At the core of our analysis is a
novel combinatorial characterization of LP decoding success, based on the
notion of a generalized matching. An interesting by-product of our analysis is
to establish the existence of ``probabilistic expansion'' in random bipartite
graphs, in which one requires only that almost every (as opposed to every) set
of a certain size expands, for sets much larger than in the classical
worst-case setting.
