Biased Random Walk on Spanning Trees of the Ladder Graph

TL;DR

This paper analyzes a biased random walk on a specially constructed spanning tree of a ladder graph, deriving explicit formulas for the walk's speed, identifying phase transitions, and confirming the Einstein relation at zero bias.

Contribution

It provides an explicit analysis of the biased random walk on a ladder graph spanning tree, including formulas for speed and phase transition points, which is novel compared to previous models.

Findings

Speed is a continuous, unimodal function of bias $eta$.

Phase transitions occur at explicit critical biases $eta_c^{(1)}$ and $eta_c^{(2)}$.

Central limit theorem holds for $eta<eta_c^{(2)}$ and Einstein relation is confirmed at $eta=1$.

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 $\beta>1$ 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.