  We show that quantum expander codes, a constant-rate family of quantum LDPC
codes, with the quasi-linear time decoding algorithm of Leverrier, Tillich and
Z\'emor can correct a constant fraction of random errors with very high
probability. This is the first construction of a constant-rate quantum LDPC
code with an efficient decoding algorithm that can correct a linear number of
random errors with a negligible failure probability. Finding codes with these
properties is also motivated by Gottesman's construction of fault tolerant
schemes with constant space overhead.
  In order to obtain this result, we study a notion of $\alpha$-percolation:
for a random subset $W$ of vertices of a given graph, we consider the size of
the largest connected $\alpha$-subset of $W$, where $X$ is an $\alpha$-subset
of $W$ if $|X \cap W| \geq \alpha |X|$.
