  We consider a specific random graph which serves as a disordered medium for a
particle performing biased random walk. Take a two-sided infinite horizontal
ladder and pick a random spanning tree with a certain edge weight $c$ for the
(vertical) rungs. Now take a random walk on that spanning tree with a bias
$\beta>1$ to the right. In contrast to other random graphs considered in the
literature (random percolation clusters, Galton-Watson trees) this one allows
for an explicit analysis based on a decomposition of the graph into independent
pieces.
  We give an explicit formula for the speed of the biased random walk as a
function of both the bias $\beta$ and the edge weight $c$. We conclude that the
speed is a continuous, unimodal function of $\beta$ that is positive if and
only if $\beta < \beta_c^{(1)}$ for an explicit critical value $\beta_c^{(1)}$
depending on $c$. In particular, the phase transition at $\beta_c^{(1)}$ is of
second order.
  We show that another second order phase transition takes place at another
critical value $\beta_c^{(2)}<\beta_c^{(1)}$ that is also explicitly known: For
$\beta<\beta_c^{(2)}$ the times the walker spends in traps have second moments
and (after subtracting the linear speed) the position fulfills a central limit
theorem. We see that $\beta_c^{(2)}$ is smaller than the value of $\beta$ which
achieves the maximal value of the speed. Finally, concerning linear response,
we confirm the Einstein relation for the unbiased model ($\beta=1$) by proving
a central limit theorem and computing the variance.
