Phase transition for the Once-reinforced random walk on $\mathbb{Z}^{d}$-like trees

Kious, Daniel; Sidoravičius, Vladas

doi:10.1214/17-aop1222

Cited by 14 publications

(15 citation statements)

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“…Remark 2. In [12], two of the authors studied the ORRW on Z d -like trees T d whose vertices have d children if they are at some generation 2 k , k ∈ N, and only one child otherwise. One can easily compute that these trees have a branching-ruin number br r (T d ) = log 2 (d) and thus recover the result from [12] using Theorem 1.…”

Section: Resultsmentioning

confidence: 99%

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The Branching‐Ruin Number and the Critical Parameter of Once‐Reinforced Random Walk on Trees

Collevecchio

Kious

Sidoravičius

2019

Comm Pure Appl Math

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The motivation for this paper is the study of the phase transition for recurrence/ transience of a class of self‐interacting random walks on trees, which includes the once‐reinforced random walk. For this purpose, we define a quantity, which we call the branching‐ruin number of a tree, which provides (in the spirit of Furstenberg [11] and Lyons [13]) a natural way to measure trees with polynomial growth. We prove that the branching‐ruin number of a tree is equal to the critical parameter for the recurrence/transience of the once‐reinforced random walk. We define a sharp and effective (i.e., computable) criterion characterizing the recurrence/transience of a larger class of self‐interacting walks on trees, providing the complete picture for their phase transition. © 2019 Wiley Periodicals, Inc.

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Section: Resultsmentioning

confidence: 99%

“…Recently, the authors in [12] provided the first example of phase transition for ORRW on Z d -like trees. It should be noted that these trees were spherically symmetric with a particular structure.…”

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confidence: 99%

The Branching‐Ruin Number and the Critical Parameter of Once‐Reinforced Random Walk on Trees

Collevecchio

Kious

Sidoravičius

2019

Comm Pure Appl Math

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“…Sidoravicius has conjectured that there is a phase transition for transience for once-reinforced simple symmetric random walk on Z d , d ≥ 3. In [18], a phase transition is shown for the (non-biased) once-reinforced random walk on Z d -like trees, which are inhomogeneous trees with polynomial growth. It should be noted that, in both [18] and the conjecture of Sidoravicius, although the underlying walk is transient, it always has zero-speed.…”

Section: Introductionmentioning

confidence: 99%

“…In [18], a phase transition is shown for the (non-biased) once-reinforced random walk on Z d -like trees, which are inhomogeneous trees with polynomial growth. It should be noted that, in both [18] and the conjecture of Sidoravicius, although the underlying walk is transient, it always has zero-speed. Here, we provide examples where the underlying walk has positive speed, but its once-reinforced counterpart is recurrent.…”

Section: Introductionmentioning

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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“…The phase transition of the once-reinforced random walk was studied in [8]. In order to see a phase transition, one needs to consider trees that grow polynomially fast (see [16]), and therefore the branching number is not the quantity that would provide a relevant information in this case. Indeed, the branching number does not allow us to distinguish among trees with polynomial growth as the branching number of any tree with sub-exponential growth is equal to 1.…”

Section: Introductionmentioning

confidence: 99%

The branching-ruin number as critical parameter of random processes on trees

Collevecchio¹,

Huynh²,

Kious³

2019

Electron. J. Probab.

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The branching-ruin number of a tree, which describes its asymptotic growth and geometry, can be seen as a polynomial version of the branching number. This quantity was defined by Collevecchio, Kious and Sidoravicius (2018) in order to understand the phase transitions of the once-reinforced random walk (ORRW) on trees. Strikingly, this number was proved to be equal to the critical parameter of ORRW on trees. In this paper, we continue the investigation of the link between the branchingruin number and the criticality of random processes on trees. First, we study random walks on random conductances on trees, when the conductances have an heavy tail at 0, parametrized by some p > 1, where 1/p is the exponent of the tail. We prove a phase transition recurrence/transience with respect to p and identify the critical parameter to be equal to the branching-ruin number of the tree. Second, we study a multi-excited random walk on trees where each vertex has M cookies and each cookie has an infinite strength towards the root. Here again, we prove a phase transition recurrence/transience and identify the critical number of cookies to be equal to the branching-ruin number of the tree, minus 1. This result extends a conjecture of Volkov (2003). Besides, we study a generalized version of this process and generalize results of Basdevant and Singh (2009).

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Phase transition for the Once-reinforced random walk on $\mathbb{Z}^{d}$-like trees

Cited by 14 publications

References 24 publications

The Branching‐Ruin Number and the Critical Parameter of Once‐Reinforced Random Walk on Trees

The Branching‐Ruin Number and the Critical Parameter of Once‐Reinforced Random Walk on Trees

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

The branching-ruin number as critical parameter of random processes on trees

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