The branching-ruin number and the critical parameter of once-reinforced random walk on trees

Collevecchio, Andrea; Kious, Daniel; Sidoravičius, Vladas

doi:10.48550/arxiv.1710.00567

Cited by 3 publications

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“…Even more is true: θ is a flow with respect to the capacities κ e if and only if it is a flow with respect to κ min 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: 96%

“…Let θ 1 be the unit flow along P. Then θ 2 = θ − α • θ 1 is also a flow from 0 to Z with number of edges satisfying θ 2 (e) = 0 less or equal than n. So we can find a measure µ 2 satisfying (9) for θ 2 instead of θ. But now the measure µ := µ 2 + α • δ P has the desired property (9) for θ.…”

Section: Random Walks and Maximal Flows For Exponential Couplingsmentioning

confidence: 99%

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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: 96%

Section: Random Walks and Maximal Flows For Exponential Couplingsmentioning

confidence: 99%

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

Bäumler¹

2019

Electron. J. Probab.

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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%

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 first example of such phase transition was provided in [9] on a particular class of trees with polynomial growth, which is in contrast with the result of Durrett, Kesten and Limic [8] who showed that the ORRW is transient on regular trees for any δ > 0 (later generalized to any supercritical tree by Collevecchio [2]). More recently, the complete picture on trees has been given in [3]: the critical parameter of ORRW on a locally finite tree is equal to its branching-ruin number, which is defined in [3] as a polynomial regime of the branching number (see [10]).…”

Section: Introduction 1general Overviewmentioning

confidence: 99%

“…We use Proposition 4.5, with ε := 1/δ η , and η = 1 8 + 1 100 . We next fix α and β such that α + β ≤ 3 4 + 1 100 . Then Proposition 4.5 shows that for δ ≥ C|Γ| 40 , with C some large constant, one has almost surely on the event…”

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confidence: 99%

Once reinforced random walk on $\mathbb{Z}\times Γ$

Kious,

Schapira,

Singh

2018

Preprint

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We revisit Vervoort's unpublished paper [16] on the once reinforced random walk, and prove that this process is recurrent on any graph of the form Z × Γ, with Γ a finite graph, for sufficiently large reinforcement parameter. We also obtain a shape theorem for the set of visited sites, and show that the fluctuations around this shape are of polynomial order. The proof involves sharp general estimates on the time spent on subgraphs of the ambiant graph which might be of independent interest.

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The branching-ruin number and the critical parameter of once-reinforced random walk on trees

Cited by 3 publications

References 0 publications

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

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

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

Once reinforced random walk on $\mathbb{Z}\times Γ$

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