Lyons‐Pemantle‐Peres Monotonicity Problem for High Biases

Arous, Gérard Ben; Fribergh, Alexander; Sidoravičius, Vladas

doi:10.1002/cpa.21505

Cited by 28 publications

(47 citation statements)

References 19 publications

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“…Here, we divide the proof of Theorem 1.12 into two main subsections. In Section 3.1, we explain the idea of the proof, which is an adaptation of an argument of Ben Arous, Fribergh and Sidoravicius [4]. We provide general statements that one should be able to apply in a variety of situations by simply checking some key requirements.…”

Section: Monotonicity Of the Speedmentioning

confidence: 99%

On the speed of once-reinforced biased random walk on trees

Collevecchio¹,

Holmes²,

Kious³

2018

Electron. J. Probab.

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We study the asymptotic behaviour of once-reinforced biased random walk (ORbRW) on Galton-Watson trees. Here the underlying (unreinforced) random walk has a bias towards or away from the root. We prove that in the setting of multiplicative once-reinforcement the ORbRW can be recurrent even when the underlying biased random walk is ballistic. We also prove that, on Galton-Watson trees without leaves, the speed is positive in the transient regime. Finally, we prove that, on regular trees, the speed of the ORbRW is monotone decreasing in the reinforcement parameter when the underlying random walk has high speed, and the reinforcement parameter is small.The proof of Theorem 1.12 is inspired by work of Ben Arous, Fribergh and Sidoravicius [4], who proved a partial monotonicity result for biased random walks on Galton-Watson trees via a nice and natural coupling. Here we needed to adapt this idea to find a coupling which works for certain self-interacting walks. We have taken some care to extract the general features of the argument (which are relatively simple) and the details of the coupling for our particular setting (which are highly non-trivial). This general method provides a coupling alternative to expansion techniques (e.g. [10]) when the self-interacting random walk has a large bias independent of the history. The modelConsider a rooted tree G = (V, E) (where V is the set of vertices and E is the set of edges), augmented by adding a parent −1 to the root , the two being connected by an edge. If two vertices ν and µ are the endpoints of the same edge, they are said to be neighbours, and this property is denoted by ν ∼ µ. The distance d(ν, µ) between any pair of vertices ν, µ, not necessarily adjacent, is the number of edges in the unique self-avoiding path connecting ν to µ. We set | −1 | = −1. For any other vertex ν, we let |ν| be the distance from ν to the root . We use ν −1 to denote the parent of ν, and (ν i ) i∈[∂(ν)] , its children, where ∂(ν) is the number of offspring of ν and [n] := {1, 2, . . . , n}. Write C ν = {ν 1 , . . . , ν ∂(ν) }. For µ ∈ V \ { −1 } we write ν < µ, if ν is an ancestor of µ, i.e. ν lies on the self-avoiding path connecting µ to −1 , and we write ν ≤ µ if ν < µ or ν = µ. If ν < µ then µ is said to be a descendant of ν.We are now about to define MAD walks, standing for maximum acts differently, in reference to their behavior on Z. Fix G = (V, E) and parameters u 1 , u 0 > 0 and define the law P G of a MAD walk X = (X n ) n∈Z + , taking values on the vertices of G (and taking nearest neighbour steps) as follows. We set X 0 = −1 . Given F n = σ(X k : k ≤ n), we let E ∅ (n) = {[X k−1 , X k ] : k ≤ n} ⊂ E denote the set of (undirected) edges crossed by time n.For ν = −1 define W n (ν, ν −1 ) = 1, (1.1)

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Section: Monotonicity Of the Speedmentioning

confidence: 99%

On the speed of once-reinforced biased random walk on trees

Collevecchio¹,

Holmes²,

Kious³

2018

Electron. J. Probab.

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“…which goes to 1 as λ → ∞. Fact 2 was proven by [BAFS14] for the biased random walk on a Galton-Watson tree without leaves (where an upper bound for λ c can be explicitly computed), the same arguments yield the analogous result for the conductance model, when the conductances are bounded away from 0 and ∞. A sketch of the proof will be given in Section 2.…”

Section: Introductionmentioning

confidence: 71%

“…Unlike in [BAFS14], we require an additional step to the right in order to decouple the environment seen by the random walker. By classical arguments, the sequence (X…”

Section: A General Couplingmentioning

confidence: 99%

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The speed of biased random walk among random conductances

Berger¹,

Gantert²,

Nagel³

2019

Ann. Inst. H. Poincaré Probab. Statist.

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We consider biased random walk among iid, uniformly elliptic conductances on Z d , and investigate the monotonicity of the velocity as a function of the bias. It is not hard to see that if the bias is large enough, the velocity is increasing as a function of the bias. Our main result is that if the disorder is small, i.e. all the conductances are close enough to each other, the velocity is always strictly increasing as a function of the bias, see Theorem 1. A crucial ingredient of the proof is a formula for the derivative of the velocity, which can be written as a covariance, see Theorem 3: it follows along the lines of the proof of the Einstein relation in [GGN]. On the other hand, we give a counterexample showing that for iid, uniformly elliptic conductances, the velocity is not always increasing as a function of the bias. More precisely, if d = 2 and if the conductances take the values 1 (with probability p) and κ (with probability 1 − p) and p is close enough to 1 and κ small enough, the velocity is not increasing as a function of the bias, see Theorem 2.

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“…In the case of Galton-Watson trees without leaves, the speed is conjectured to be increasing as a function of the bias. This conjecture is proved for large enough bias by Ben Arous, Fribergh and Sidoravicius in [7]. Aïdékon gave in [1] a formula for the speed of biased random walks on Galton-Watson trees, which allows to deduce monotonicity for a larger (but not the full) range of parameters.…”

Section: Introductionmentioning

confidence: 96%

Regularity of the Speed of Biased Random Walk in a One-Dimensional Percolation Model

2018

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We consider biased random walks on the infinite cluster of a conditional bond percolation model on the infinite ladder graph. Axelson-Fisk and Häggström established for this model a phase transition for the asymptotic linear speed v of the walk. Namely, there exists some critical value λ c > 0 such that v > 0 if λ ∈ (0, λ c ) and v = 0 if λ ≥ λ c . We show that the speed v is continuous in λ on (0, ∞) and differentiable on (0, λ c /2). Moreover, we characterize the derivative as a covariance. For the proof of the differentiability of v on (0, λ c /2), we require and prove a central limit theorem for the biased random walk. Additionally, we prove that the central limit theorem fails to hold for λ ≥ λ c /2.

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Lyons‐Pemantle‐Peres Monotonicity Problem for High Biases

Abstract: The speed v.ˇ/ of aˇ-biased random walk on a Galton-Watson tree without leaves is increasing forˇ 1160.

Cited by 28 publications

References 19 publications

On the speed of once-reinforced biased random walk on trees

On the speed of once-reinforced biased random walk on trees

The speed of biased random walk among random conductances

Regularity of the Speed of Biased Random Walk in a One-Dimensional Percolation Model

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