Power-law decay of weights and recurrence of the two-dimensional VRJP

Kozma, Gady; Peled, Ron

doi:10.1214/21-ejp639

Cited by 6 publications

(5 citation statements)

References 24 publications

(36 reference statements)

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“…This basic (and somewhat vague) approach has been key in many proofs of the Mermin-Wagner theorem in statistical physics, including [18,33,19,39,41,34], [22,Theorem 9.2] and [38, Section 2.6]), whence the name, but has also been used in other contexts, e.g. in [42,12,30,21]. Our treatment here draws inspiration from [39,41,34,30] and has the benefit of providing Gaussian lower bounds on the tail probabilities.…”

Section: )mentioning

confidence: 99%

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Coalescence of geodesics and the BKS midpoint problem in planar first-passage percolation

Dembin¹,

Elboim²,

Peled³

2022

Preprint

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We consider first-passage percolation on Z 2 with independent and identically distributed weights whose common distribution is absolutely continuous with a finite exponential moment. Under the assumption that the limit shape has more than 32 extreme points, we prove that geodesics with nearby starting and ending points have significant overlap, coalescing on all but small portions near their endpoints. The statement is quantified, with power-law dependence of the involved quantities on the length of the geodesics.The result leads to a quantitative resolution of the Benjamini-Kalai-Schramm midpoint problem. It is shown that the probability that the geodesic between two given points passes through a given edge is smaller than a power of the distance between the points and the edge.We further prove that the limit shape assumption is satisfied for a specific family of distributions.

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Section: )mentioning

confidence: 99%

“…in [42,12,30,21]. Our treatment here draws inspiration from [39,41,34,30] and has the benefit of providing Gaussian lower bounds on the tail probabilities.…”

Section: )mentioning

confidence: 99%

Coalescence of geodesics and the BKS midpoint problem in planar first-passage percolation

Dembin¹,

Elboim²,

Peled³

2022

Preprint

Self Cite

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“…In this case, when d ∈ {1, 2}, the VRJP is always recurrent. (See [SZ19], [Sab21] and [KP21].) On the contrary, when d ≥ 3, Sabot and Tarrès proved in [ST15] that the time-changed VRJP is recurrent for small W and that it is transient for large W .…”

Section: Introduction and First Definitionsmentioning

confidence: 99%

About the asymptotic behaviour of the martingale associated with the Vertex Reinforced Jump Process on trees and $\mathbb{Z}^d$

Rapenne¹

2022

Preprint

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We study the asymptotic behaviour of the martingale (ψ n (o)) n∈N associated with the Vertex Reinforced Jump Process (VRJP). We show that it is bounded in L p for every p > 1 on trees and uniformly integrable on Z d in all the transient phase of the VRJP. Moreover, when the VRJP is recurrent on trees, we have good estimates on the moments of ψ n (o) and we can compute the exact decreasing rate τ such that n −1 ln(ψ n (o)) ∼ −τ almost surely where τ is related to standard quantities for branching random walks. Besides, on trees, at the critical point, we show that n −1/3 ln(ψ n (o)) ∼ −ρ c almost surely where ρ c can be computed explicitely. Furthermore, at the critical point, we prove that the discrete process associated with the VRJP is a mixture of positive recurrent Markov chains. Our proofs use properties of the β-potential associated with the VRJP and techniques coming from the domain of branching random walks.

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“…The result holds for any temperature in dimension one. In two dimensions, polynomial decay of the correlations was shown for all temperatures independently by Kozma and Peled (2021) and Sabot (2021), using different tools.…”

Section: Introductionmentioning

confidence: 99%

The non-linear supersymmetric hyperbolic sigma model on a complete graph with hierarchical interactions

Disertori¹,

Merkl²,

Rolles³

2022

ALEA

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We study the non-linear supersymmetric hyperbolic sigma model H 2|2 on a complete graph with hierarchical interactions. For interactions which do not decrease too fast in the hierarchical distance, we prove tightness of certain spin variables in horospherical coordinates, uniformly in the pinning and in the size of the graph. The proof relies on a reduction to an effective H 2|2 -model; its size is logarithmic in the size of the original model.

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Power-law decay of weights and recurrence of the two-dimensional VRJP

Cited by 6 publications

References 24 publications

Coalescence of geodesics and the BKS midpoint problem in planar first-passage percolation

Coalescence of geodesics and the BKS midpoint problem in planar first-passage percolation

About the asymptotic behaviour of the martingale associated with the Vertex Reinforced Jump Process on trees and $\mathbb{Z}^d$

The non-linear supersymmetric hyperbolic sigma model on a complete graph with hierarchical interactions

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