# Randomly biased walks on subcritical trees

@article{BenArous2011RandomlyBW, title={Randomly biased walks on subcritical trees}, author={G{\'e}rard Ben Arous and Alan Hammond}, journal={Communications on Pure and Applied Mathematics}, year={2011}, volume={65} }

As a model of trapping by biased motion in random structure, we study the time taken for a biased random walk to return to the root of a subcritical Galton‐Watson tree. We do so for trees in which these biases are randomly chosen, independently for distinct edges, according to a law that satisfies a logarithmic nonlattice condition. The mean return time of the walk is in essence given by the total conductance of the tree. We determine the asymptotic decay of this total conductance, finding it…

## 16 Citations

Stable limit laws for randomly biased walks on supercritical trees

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- MathematicsElectronic Journal of Probability
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The potential energy of biased random walks on trees

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Biased random walks on supercritical Galton--Watson trees are introduced and studied in depth by Lyons (1990) and Lyons, Pemantle and Peres (1996). We investigate the slow regime, in which case the…

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- Mathematics, PhysicsProbability theory and related fields
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Biasful random walk on subcritical and supercritical Galton–Watson trees conditioned to survive in the transient, sub-ballistic regime is studied and new trapping phenomena for the walk on the subcritical tree which, in this case, always yield sub- ballisticity.

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Consider a class of null-recurrent randomly biased walks on a supercritical Galton–Watson tree. We obtain the scaling limits of the local times and the quenched local probability for the biased walk…

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- Mathematics
- 2013

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We consider biased random walks in positive random conductances on the d-dimensional lattice in the zero-speed regime and study their scaling limits. We obtain a functional Law of Large Numbers for…

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## References

SHOWING 1-10 OF 37 REFERENCES

Stable limit laws for randomly biased walks on supercritical trees

- Mathematics
- 2011

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…

Almost sure convergence for stochastically biased random walks on trees

- Mathematics
- 2012

We are interested in the biased random walk on a supercritical Galton–Watson tree in the sense of Lyons (Ann. Probab. 18:931–958, 1990) and Lyons, Pemantle and Peres (Probab. Theory Relat. Fields…

SCALING LIMIT FOR TRAP MODELS ON

- Mathematics
- 2007

We give the " quenched " scaling limit of Bouchaud's trap model in d ≥ 2. This scaling limit is the fractional-kinetics process, that is the time change of a d-dimensional Brownian motion by the…

Biased random walks on Galton–Watson trees

- Mathematics
- 1996

Summary. We consider random walks with a bias toward the root on the family tree T of a supercritical Galton–Watson branching process and show that the speed is positive whenever the walk is…

Phase Transition for the Speed of the Biased Random Walk on the Supercritical Percolation Cluster

- Mathematics
- 2011

We prove the sharpness of the phase transition for the speed in biased random walk on the supercritical percolation cluster on ℤd. That is, for each d ≥ 2, and for any supercritical parameter p > pc,…

Transient random walks in random environment on a Galton-Watson tree

- Mathematics
- 2008

We consider a transient random walk (Xn) in random environment on a Galton–Watson tree. Under fairly general assumptions, we give a sharp and explicit criterion for the asymptotic speed to be…

Random trees and applications

- Mathematics, Computer Science
- 2005

An introduction to the theory of the Brownian snake, which combines the genealogical structure of random real trees with independent spatial motions, and some applications to a class of semilinear partial differential equations.

UNSOLVED PROBLEMS CONCERNING RANDOM WALKS ON TREES

- Mathematics
- 1997

We state some unsolved problems and describe relevant examples concerning random walks on trees. Most of the problems involve the behavior of random walks with drift: e.g., is the speed on…

Limit laws for transient random walks in random environment on $\z$

- Mathematics
- 2007

We consider transient random walks in random environment on $\z$ with zero asymptotic speed. A classical result of Kesten, Kozlov and Spitzer says that the hitting time of the level $n$ converges in…

Drift and Trapping in Biased Diffusion on Disordered Lattices

- Physics
- 1998

We re-examine the theory of transition from drift to no-drift in biased diffusion on percolation networks. We argue that for the bias field B equal to the critical value Bc, the average velocity at…