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

Collevecchio, Andrea; Kious, Daniel; Sidoravičius, Vladas

doi:10.1002/cpa.21860

Cited by 9 publications

(22 citation statements)

References 25 publications

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“…Let (ω n ) n∈N be a summable sequence of positive real numbers. Then, as already noted in ( [6], Using the equivalences proven in Theorem 2.1 this also follows from a special case of [9], but we give a different proof, as we will use the same technique in a slightly different setting again at a later point.…”

Section: A Tree With Two Different Coupling Distributionsmentioning

confidence: 77%

Uniqueness and non-uniqueness for spin-glass ground states on trees

Bäumler¹

2019

Electron. J. Probab.

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We consider a spin glass at temperature T = 0 where the underlying graph is a locally finite tree. We prove for a wide range of coupling distributions that uniqueness of ground states is equivalent to the maximal flow from any vertex to ∞ (where each edge e has capacity |J e |) being equal to zero which is equivalent to recurrence of the simple random walk on the tree.Let G = (V, E) be a locally finite graph, for a given finite set B ⊂ V define E(B) as the set of edges with at least one end in B. For any finite setIn the Edwards-Anderson spin glass model [8] one considers nearest neighbor interactions and the case where the J xy ′ s, also called the couplings, are i.i.d. random variables. The distribution of a single coupling will be denoted by ν, throughout we assume that ν({0}) = 0. With ν E we denote the distribution of (J xy , (x, y) ∈ E). We call an edge e = (x, y) ∈ E satisfied for a configuration σ if J xy σ x σ y > 0, if J xy σ x σ y < 0 we call it unsatisfied. Note that ν E (∃e ∈ E such that J xy = 0) = 0. Ground states are local minima of the Hamiltonian defined in (1), i.e. the Hamiltonian (1) can not be lowered by flipping the spins for some finite B ⊂ V . This means

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Section: A Tree With Two Different Coupling Distributionsmentioning

confidence: 77%

Uniqueness and non-uniqueness for spin-glass ground states on trees

Bäumler¹

2019

Electron. J. Probab.

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“…Here, we define a construction that is closely related to the ones introduced in [5] and [6]. This construction allows to decouple the behaviour of the process on subtrees, even when the process is transient.…”

Section: Extension Processesmentioning

confidence: 99%

“…As it was done in [6], we are now going to define a family of coupled processes on the subtrees of G. For any rooted subtree…”

Section: Extension Processesmentioning

confidence: 99%

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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“…For instance, for trees that are "well-behaved" (such as spherically symmetric trees) and whose spheres of diameter n have size m n , the branching number is equal to m. This description is actually not accurate as some trees have a peculiar geometry, and the size of their spheres is not a good indicator of their asymptotic complexity. 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.…”

Section: Introductionmentioning

confidence: 99%

“…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. In [8], it was proved that the critical parameter for the once-reinforced random walk on trees is equal to the branching-ruin number of the tree (see (2.2)). The branching-ruin number of a tree is best described as the polynomial version of the branching number: if a well-behaved tree has spheres of size n b , then the branching-ruin number of this tree is b.…”

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

Cited by 9 publications

References 25 publications

Uniqueness and non-uniqueness for spin-glass ground states on trees

Uniqueness and non-uniqueness for spin-glass ground states on trees

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

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

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