Biased Random Walk on Spanning Trees of the Ladder Graph

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