Biased random walks on Galton–Watson trees with leaves

“…On Galton-Watson trees with leaves (see [2,3,10,13]) or on supercritical percolation clusters (see [5,8,16]), the speed is certainly not increasing since it eventually vanishes. In these models, the slowdown of the walk can be explained by the presence of dead ends in the environment, which act as powerful traps.…”

Section: Introductionmentioning

confidence: 99%

Lyons‐Pemantle‐Peres Monotonicity Problem for High Biases

Arous

Fribergh

Sidoravičius

2014

Comm Pure Appl Math

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“…Instead, the random environment arising in the percolation model creates traps which become stronger as the bias is increased, so that when the bias is set above a certain critical value, the speed of the biased random walk is zero [5,9,16]. This phenomenon, which has also been observed for the biased random walks on supercritical Galton-Watson trees [4,14] and a one-dimensional percolation model [1], is of physical significance, as it helps to explain how a particle could in some circumstances actually move more slowly when the strength of an external field, such as gravity, is greater [3].…”

Section: Introductionmentioning

confidence: 69%

“…Moreover, we can define an annealed law for X started from 0 by integrating out the environment, similarly to (4). To connect this model with the biased random walk on the random path introduced above, we suppose that the transition probabilities are defined by setting…”

Section: Biased Random Walk On a Self-avoiding Random Pathmentioning

confidence: 99%

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Slow movement of a random walk on the range of a random walk in the presence of an external field

Croydon

2012

Probab. Theory Relat. Fields

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In this article, a localisation result is proved for the biased random walk on the range of a simple random walk in high dimensions (d ≥ 5). This demonstrates that, unlike in the supercritical percolation setting, a slowdown effect occurs as soon as a non-trivial bias is introduced. The proof applies a decomposition of the underlying simple random walk path at its cut-times to relate the associated biased random walk to a one-dimensional random walk in a random environment in Sinai's regime. Via this approach, a corresponding aging result is also proved.

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Randomly biased walks on subcritical trees

Arous

Hammond

2012

Comm Pure Appl Math

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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 non-lattice 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 to have a pure power-law decay. In the case of the conductance associated to a single vertex at maximal depth in the tree, this asymptotic decay may be analysed by the classical defective renewal theorem, due to the non-lattice edge-bias assumption. However, the derivation of the decay for total conductance requires computing an additional constant multiple outside the power-law that allows for the contribution of all vertices close to the base of the tree. This computation entails a detailed study of a convenient decomposition of the tree, under conditioning on the tree having high total conductance. As such, our principal conclusion may be viewed as a development of renewal theory in the context of random environments.For randomly biased random walk on a supercritical Galton-Watson tree with positive extinction probability, our main results may be regarded as a description of the slowdown mechanism caused by the presence of subcritical trees adjacent to the backbone that may act as traps that detain the walker. Indeed, this conclusion is exploited in [18] to obtain a stable limiting law for walker displacement in such a tree.

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Biased random walks on Galton–Watson trees with leaves

Cited by 46 publications

References 19 publications

Lyons‐Pemantle‐Peres Monotonicity Problem for High Biases

Lyons‐Pemantle‐Peres Monotonicity Problem for High Biases

Slow movement of a random walk on the range of a random walk in the presence of an external field

Randomly biased walks on subcritical trees

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