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

Aïdékon, Élie

doi:10.1007/s00440-007-0114-x

Cited by 33 publications

(125 citation statements)

References 13 publications

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“…Recall the definition of ω(ν, νi) given in (1.1), and that for LERRW on the b-regular tree and for |ν| ≥ 1, we have ω(ν, νi) is distributed as Beta(1/2, (b + 1)/2), while ω(ν, ν −1 ) is a Beta(1, b/2). The reasoning presented in this section follows closely the one given in section 7 in [1].…”

Section: Linearly Edge-reinforced Random Walksmentioning

confidence: 85%

“…Proof. This proof is inspired by the proof of Lemma 2.2 in [1]. We include the steps for completeness.…”

Section: Finite Second Moment Between Cut Timesmentioning

confidence: 99%

“…Following [1] (proof of Lemma 2.2), by the i.i.d. structure of the environment, we have P(E n ) = P(E 1 ) n .…”

Section: Finite Second Moment Between Cut Timesmentioning

confidence: 99%

“…random environment to study LERRW. LERRW on the binary tree is transient and has positive speed, even though does not satisfy (1.3) (see [1]). Our result is the following and improves Theorem 3 of [4].…”

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confidence: 99%

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Functional central limit theorem for random walks in random environment defined on regular trees

Collevecchio

Takei

Uematsu

2020

Stochastic Processes and their Applications

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We study Random Walks in an i.i.d. Random Environment (RWRE) defined on b-regular trees. We prove a functional central limit theorem (FCLT) for transient processes, under a moment condition on the environment. We emphasize that we make no uniform ellipticity assumptions. Our approach relies on regenerative levels, i.e. levels that are visited exactly once. On the way, we prove that the distance between consecutive regenerative levels have a geometrically decaying tail. In the second part of this paper, we apply our results to Linearly Edge-Reinforced Random Walk (LERRW) to prove FCLT when the process is defined on b-regular trees, with b ≥ 4, substantially improving the results of the first author (see Theorem 3 of [4]). introductionRandom Walk in Random Environment (RWRE) is a class of self-interacting processes that attracted much attention from probabilists since the seminal work of Kesten, Kozlov, and Spitzer [10] and Solomon [15], in the 70's. It seems that the initial motivation behind this class of process was related to problems in biology, crystallography and metal physics. The interest in this field grew substantially, and we refer to [3] and [16] for an overview of this beautiful subject.We study random walks in an i.i.d. random environment defined on b-regular trees. We provide a functional central limit theorem (FCLT) for processes that are transient, assuming a moment condition on the environment. Our approach relies on regenerative levels, i.e. levels that are visited exactly once. On the way, we prove that the space between these regenerative levels have a geometrically decaying tail. We emphasize that we make no uniform ellipticity assumptions.

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Section: Linearly Edge-reinforced Random Walksmentioning

confidence: 85%

“…Proof. This proof is inspired by the proof of Lemma 2.2 in [1]. We include the steps for completeness.…”

Section: Finite Second Moment Between Cut Timesmentioning

confidence: 99%

“…Following [1] (proof of Lemma 2.2), by the i.i.d. structure of the environment, we have P(E n ) = P(E 1 ) n .…”

Section: Finite Second Moment Between Cut Timesmentioning

confidence: 99%

mentioning

confidence: 99%

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Functional central limit theorem for random walks in random environment defined on regular trees

Collevecchio

Takei

Uematsu

2020

Stochastic Processes and their Applications

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“…In particular there are several regimes for both recurrent and transient cases. A complete classification for the recurrent cases is given by G. Faraud [Far11] (for the transient cases, see E. Aidekon [Aïd08]). It can be determined from the fluctuations of log-Laplace transform ψ(s) := log E |z|=1 e −sV (z) as resumed in Figure 1.…”

Section: Introductionmentioning

confidence: 99%

The heavy range of randomly biased walks on trees

Andreoletti

Diel²

2020

Stochastic Processes and their Applications

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We focus on recurrent random walks in random environment (RWRE) on Galton-Watson trees. The range of these walks, that is the number of sites visited at some fixed time, has been studied in three different papers [AC18], [AdR17] and [dR16]. Here we study the heavy range: the number of edges visited at least α times for some integer α. The asymptotic behavior of this process when α is a power of the number of steps of the walk is given for all the recurrent cases. It turns out that this heavy range plays a crucial role in the rate of convergence of an estimator of the environment from a single trajectory of the RWRE.MSC : Primary 60K37, 60J80; Secondary 62G05.

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

Cited by 33 publications

References 13 publications

Functional central limit theorem for random walks in random environment defined on regular trees

Functional central limit theorem for random walks in random environment defined on regular trees

The heavy range of randomly biased walks on trees

Randomly biased walks on subcritical trees

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