Abstract
AbstractWe 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.
Funder
Technische Universität München
Publisher
Springer Science and Business Media LLC
Subject
Mathematical Physics,Statistical and Nonlinear Physics
Reference26 articles.
1. Aïdékon, E.: Speed of the biased random walk on a Galton-Watson tree. Probab. Theory Relat. Fields 159(3–4), 597–617 (2014)
2. Balakrishnan, V., Van den Broeck, C.: Transport properties on a random comb. Phys. A 217, 1–21 (1995)
3. Barma, M., Dhar, D.: Directed diffusion in a percolation network. J. Phys. C 16(8), 1451 (1983)
4. Ben Arous, G., Cabezas, M., Černý, J., Royfman, R.: Randomly trapped random walks. Ann. Probab. 43(5), 2405–2457 (2015)
5. Ben Arous, G., Fribergh, A.: Biased random walks on random graphs. In: Probability and Statistical Physics in St. Petersburg, vol. 91 of Proc. Sympos. Pure Math., pp. 99–153. Amer. Math. Soc., Providence (2016)