## The Annals of Probability

### Stable limit laws for randomly biased walks on supercritical trees

Alan Hammond

#### Abstract

We consider a random walk on a supercritical Galton–Watson tree with leaves, where the transition probabilities of the walk are determined by biases that are randomly assigned to the edges of the tree. The biases are chosen independently on distinct edges, each one according to a given law that satisfies a logarithmic nonlattice condition. We determine the condition under which the walk is sub-ballistic, and, in the sub-ballistic regime, we find a formula for the exponent $\gamma \in(0,1)$ such that the distance $\vert X(n)\vert$ moved by the walk in time $n$ is of the order of $n^{\gamma }$. We prove a stable limiting law for walker distance at late time, proving that the rescaled walk $n^{-\gamma }\vert X(n)\vert$ converges in distribution to an explicitly identified function of the stable law of index $\gamma$.

This paper is a counterpart to Ben Arous et al. [Ann. Probab. 40 (2012) 280–338], in which it is proved that, in the model where the biases on edges are taken to be a given constant, there is a logarithmic periodicity effect that prevents the existence of a stable limit law for scaled walker displacement. It is randomization of edge-biases that is responsible for the emergence of the stable limit in the present article, while also introducing further correlations into the model in comparison with the constant bias case. The derivation requires the development of a detailed understanding of trap geometry and the interplay between traps and backbone. The paper may be considered as a sequel to Ben Arous and Hammond [Comm. Pure Appl. Math. 65 (2012) 1481–1527], since it makes use of a result on the regular tail of the total conductance of a randomly biased subcritical Galton–Watson tree.

#### Article information

Source
Ann. Probab., Volume 41, Number 3A (2013), 1694-1766.

Dates
First available in Project Euclid: 29 April 2013

https://projecteuclid.org/euclid.aop/1367241510

Digital Object Identifier
doi:10.1214/12-AOP752

Mathematical Reviews number (MathSciNet)
MR3098688

Zentralblatt MATH identifier
1304.60110

#### Citation

Hammond, Alan. Stable limit laws for randomly biased walks on supercritical trees. Ann. Probab. 41 (2013), no. 3A, 1694--1766. doi:10.1214/12-AOP752. https://projecteuclid.org/euclid.aop/1367241510

#### References

• [1] Aidékon, E. (2008). Transient random walks in random environment on a Galton–Watson tree. Probab. Theory Related Fields 142 525–559.
• [2] Barlow, M. Random walks on graphs: A brief introduction. Available at http://www.math.ubc.ca/~barlow/cornell/cnotes.pdf.
• [3] Ben Arous, G., Fribergh, A., Gantert, N. and Hammond, A. (2012). Biased random walks on Galton–Watson trees with leaves. Ann. Probab. 40 280–338.
• [4] Ben Arous, G. and Černý, J. (2007). Scaling limit for trap models on $\mathbb{Z}^{d}$. Ann. Probab. 35 2356–2384.
• [5] Ben Arous, G. and Hammond, A. (2012). Randomly biased walks on subcritical trees. Comm. Pure Appl. Math. 65 1481–1527.
• [6] Berger, N., Gantert, N. and Peres, Y. (2003). The speed of biased random walk on percolation clusters. Probab. Theory Related Fields 126 221–242.
• [7] Bottger, H. and Bryksin, V. (1982). Hopping conductivity in ordered and disordered systems (III). Physica Status Solidi (b) 113 9–49.
• [8] Bouchaud, J. P. (1992). Weak ergodicity breaking and aging in disordered-systems. J. Phys. (France) I 2 1705–1713.
• [9] Chandra, A. K., Raghavan, P., Ruzzo, W. L., Smolensky, R. and Tiwari, P. (1996/97). The electrical resistance of a graph captures its commute and cover times. Comput. Complexity 6 312–340.
• [10] Dhar, D. (1984). Diffusion and drift on percolation networks in an external field. J. Phys. A 17 257–259.
• [11] Dhar, D. and Stauffer, D. (1998). Drift and trapping in biased diffusion on disordered lattices. Internat. J. Modern Phys. C 9 349–355.
• [12] Durrett, R. (1996). Probability: Theory and Examples, 2nd ed. Duxbury Press, Belmont, CA.
• [13] Enriquez, N., Sabot, C. and Zindy, O. (2009). Limit laws for transient random walks in random environment on $\mathbb{Z}$. Ann. Inst. Fourier (Grenoble) 59 2469–2508.
• [14] Enriquez, N., Sabot, C. and Zindy, O. (2009). A probabilistic representation of constants in Kesten’s renewal theorem. Probab. Theory Related Fields 144 581–613.
• [15] Faraud, G., Hu, Y. and Shi, Z. (2012). Almost sure convergence for stochastically biased random walks on trees. Probab. Theory Related Fields 154 621–660.
• [16] Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. II, 2nd ed. Wiley, New York.
• [17] Fribergh, A. and Hammond, A. (2013). Phase transition for the speed of the biased random walk on the supercritical percolation cluster. Comm. Pure Appl. Math. To appear. Available at arXiv:1103.1371.
• [18] Hammond, A. (2012). Stable limit laws for randomly biased walks on supercritical trees. Preprint. Available at arXiv:1101.4043.
• [19] Kesten, H., Kozlov, M. V. and Spitzer, F. (1975). A limit law for random walk in a random environment. Compos. Math. 30 145–168.
• [20] Le Gall, J.-F. (2005). Random trees and applications. Probab. Surv. 2 245–311.
• [21] Lyons, R. and Pemantle, R. (1992). Random walk in a random environment and first-passage percolation on trees. Ann. Probab. 20 125–136.
• [22] Lyons, R., Pemantle, R. and Peres, Y. (1996). Biased random walks on Galton–Watson trees. Probab. Theory Related Fields 106 249–264.
• [23] Lyons, R. and Peres, Y. Probability on trees and networks. Available at http://mypage.iu.edu/~rdlyons/prbtree/book.pdf.
• [24] Stauffer, D. and Sornette, D. (1998). Log-periodic oscillations for biased diffusion on random lattice. Phys. A 252 271–277.
• [25] Sznitman, A.-S. (2003). On the anisotropic walk on the supercritical percolation cluster. Comm. Math. Phys. 240 123–148.