Biased Random Walk on Spanning Trees of the Ladder Graph

Nina Gantert, Achim Klenke

Published 2022-10-14Version 1

Consider 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. We give an explicit formula for the speed of that random walk and show that it 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 $c$. We also show a central limit theorem for $\beta< \beta_c^{(2)} $, for an explicit critical value $\beta_c^{(2)}$. We see that $\beta_c^{(2)}$ is smaller than the value of $\beta$ which achieves the maximal value of the speed. Finally, we confirm the Einstein relation for the unbiased model ($\beta=1$) by proving a Central Limit Theorem and computing the variance.