Biased Random Walk on Spanning Trees of the Ladder Graph

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.