## Electronic Communications in Probability

### Random walk on the randomly-oriented Manhattan lattice

#### Abstract

In the randomly-oriented Manhattan lattice, every line in $\mathbb{Z} ^d$ is assigned a uniform random direction. We consider the directed graph whose vertex set is $\mathbb{Z} ^d$ and whose edges connect nearest neighbours, but only in the direction fixed by the line orientations. Random walk on this directed graph chooses uniformly from the $d$ legal neighbours at each step. We prove that this walk is superdiffusive in two and three dimensions. The model is diffusive in four and more dimensions.

#### Article information

Source
Electron. Commun. Probab., Volume 23 (2018), paper no. 43, 11 pp.

Dates
Accepted: 19 June 2018
First available in Project Euclid: 25 July 2018

https://projecteuclid.org/euclid.ecp/1532505674

Digital Object Identifier
doi:10.1214/18-ECP144

Mathematical Reviews number (MathSciNet)
MR3841404

Zentralblatt MATH identifier
1397.82046

#### Citation

Ledger, Sean; Tóth, Bálint; Valkó, Benedek. Random walk on the randomly-oriented Manhattan lattice. Electron. Commun. Probab. 23 (2018), paper no. 43, 11 pp. doi:10.1214/18-ECP144. https://projecteuclid.org/euclid.ecp/1532505674

#### References

• [1] N. Guillotin-Plantard and A. Le Ny. Transient random walks on 2D oriented lattices. Theo. Probab. Appl., 52(4):699–711, 2008.
• [2] N. Guillotin-Plantard and A. Le Ny. A functional limit theorem for a 2d-random walk with dependent marginals. Electronic Communications in Probability, 13(34):337–351, 2008.
• [3] T. Komorowski and S. Olla. On the superdiffusive behaviour of passive tracer with a Gaussian drift. J. Stat. Phys., 108:647–668, 2002.
• [4] G. Kozma and B. Tóth. Central limit theorem for random walks in doubly stochastic random environment: $\mathscr{H} _{-1}$ suffices. Annals of Probability, 45:4307–4347, 2017.
• [5] C. Landim, J. Quastel, M. Salmhoffer and H.-T. Yau. Superdiffusivity of asymmetric exclusion process in dimensions one and two. Communications in Mathematical Physics, 244:455–481, 2004.
• [6] G. Matheron and G. de Marsily. Is transport in porous media always diffusive? A counterexample. Water Resources Res., 16:901–907, 1980. https://doi.org/10.1029/WR016i005p00901
• [7] S. Sethuraman. Central limit theorems for additive functionals of the simple exclusion process Annals of Probability, 28:277–302, 2000.
• [8] P. Tarrès, B. Tóth and B. Valkó. Diffusivity bounds for 1d Brownian polymers Annals of Probability, 40:695–713, 2012.
• [9] B. Tóth. Quenched central limit theorem for random walks in doubly stochastic random environment. To appear: Annals of Probability. arXiv:1704.06072, 2018.
• [10] B. Tóth and B. Valkó. Superdiffusive bounds on self-repellent Brownian polymers and diffusion in the curl of the Gaussian free field in $d=2$. Journal of Statistical Physics, 147:113–131, 2012.
• [11] H.-T. Yau. $(\log t)^{2/3}$ law of the two dimensional asymmetric simple exclusion process. Annals of Mathematics, 159:377–405, 2004.