Biased Random Walk on Spanning Trees of the Ladder Graph

Abstract

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 $$\beta >1$$ β > 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 $$\beta $$ β and the edge weight c. We conclude that the speed is a continuous, unimodal function of $$\beta $$ β that is positive if and only if $$\beta < \beta _c^{(1)}$$ β < β c ( 1 ) for an explicit critical value $$\beta _c^{(1)}$$ β c ( 1 ) depending on c. In particular, the phase transition at $$\beta _c^{(1)}$$ β c ( 1 ) is of second order. We show that another second order phase transition takes place at another critical value $$\beta _c^{(2)}<\beta _c^{(1)}$$ β c ( 2 ) < β c ( 1 ) that is also explicitly known: For $$\beta <\beta _c^{(2)}$$ β < β 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 $$\beta _c^{(2)}$$ β c ( 2 ) is smaller than the value of $$\beta $$ β which achieves the maximal value of the speed. Finally, concerning linear response, we confirm the Einstein relation for the unbiased model ($$\beta =1$$ β = 1 ) by proving a central limit theorem and computing the variance.