  We consider biased random walks in a one-dimensional percolation model. This
model goes back to Axelson-Fisk and H\"aggstr\"om and exhibits the same phase
transition as biased random walk on the infinite cluster of supercritical
Bernoulli bond percolation on $\mathbb{Z}^d$, namely, for some critical value
$\lambda_{\mathrm{c}} >0$ of the bias, it holds that the asymptotic linear
speed $\overline{\mathrm{v}}$ of the walk is strictly positive if the bias
$\lambda$ is strictly smaller than $\lambda_{\mathrm{c}}$, whereas
$\overline{\mathrm{v}}=0$ if $\lambda \geq \lambda_{\mathrm{c}}$.
  We show that at the critical bias $\lambda = \lambda_{\mathrm{c}}$, the
displacement of the random walk from the origin is of order $n/\log n$. This is
in accordance with simulation results by Dhar and Stauffer for biased random
walk on the infinite cluster of supercritical bond percolation on
$\mathbb{Z}^d$.
  Our result is based on fine estimates for the tails of suitable regeneration
times. As a by-product of these estimates we also obtain the order of
fluctuations of the walk in the sub-ballistic and in the ballistic,
nondiffusive phase.
