Local stationarity in exponential last-passage percolation

Balázs, Márton; Busani, Ofer; Seppäläinen, Timo

doi:10.1007/s00440-021-01035-7

Cited by 19 publications

(3 citation statements)

References 23 publications

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“…Remark 4.3. Let us compare Theorem 4.2 with a related previous result from [7]. Let ą 0, N P Z ą0 and R N " Z 2 ą0 X r0, N 2{3 s 2 .…”

Section: Applications Of Exit Point Boundsmentioning

confidence: 84%

Optimal-order exit point bounds in exponential last-passage percolation via the coupling technique

Emrah¹,

Janjigian²,

Seppäläinen³

2021

Preprint

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We develop a new probabilistic method for deriving deviation estimates in directed planar polymer and percolation models. The key estimates are for exit points of geodesics as they cross transversal down-right boundaries. These bounds are of optimal cubic-exponential order. We derive them in the context of last-passage percolation with exponential weights with near-stationary boundary conditions. As a result, the probabilistic coupling method is empowered to treat a variety of problems optimally, which could previously be achieved only via inputs from integrable probability. As applications in the bulk setting, we obtain upper bounds of cubic-exponential order for transversal fluctuations of geodesics, and cube-root upper bounds with a logarithmic correction for distributional Busemann limits and competition interface limits. Several other applications are already in the literature.

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“…Remark 4.3. Let us compare Theorem 4.2 with a related previous result from [7]. Let ą 0, N P Z ą0 and R N " Z 2 ą0 X r0, N 2{3 s 2 .…”

Section: Applications Of Exit Point Boundsmentioning

confidence: 84%

Optimal-order exit point bounds in exponential last-passage percolation via the coupling technique

Emrah¹,

Janjigian²,

Seppäläinen³

2021

Preprint

Self Cite

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“…A second comparison result is obtained when the randomness on the diagonal as well on the horizontal axis are coupled in such a way that there is a certain ordering, see Lemma B.1 of [10] for the analogue result in the full-space geometry. Proposition 3.3.…”

Section: Comparison Results For Half-space Lppmentioning

confidence: 99%

Time-time covariance for last passage percolation in half-space

Ferrari¹,

Occelli²

2022

Preprint

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This article studies several properties of the half-space last passage percolation, in particular the two-time covariance. We show that, when the two end-points are at small macroscopic distance, then the first order correction to the covariance for the point-to-point model is the same as the one of the stationary model. In order to obtain the result, we first derive comparison inequalities of the last passage increments for different models. This is used to prove tightness of the point-to-point process as well as localization of the geodesics. Unlike for the full-space case, for half-space we have to overcome the difficulty that the point-to-point model in half-space with generic start and end points is not known.

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“…Basu, Sarkar and Sly [9] proved that 2/3 is the right exponent by showing that geodesics at distance Cn 2/3 will coalesce with probability tending to 1 as C ↓ 0. Zhang [55], and independently Balázs, Busani and Seppäläinen [6] and Seppäläinen and Shen [45] improved the quantitative bounds and estimated the probability of coalescence up to constants as C → 0 and as C → ∞. See also [46,8,47,48,10] for the study of infinite geodesics in exactly-solvable models.…”

Section: Introductionmentioning

confidence: 97%

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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Local stationarity in exponential last-passage percolation

Abstract: We consider point-to-point last-passage times to every vertex in a neighbourhood of size $$\delta N^{\nicefrac {2}{3}}$$ δ N 2 3 at distance N from the starting point. The increments of the last-passage times in this neighbourhood are shown to be jointly… Show more

Cited by 19 publications

References 23 publications

Optimal-order exit point bounds in exponential last-passage percolation via the coupling technique

Optimal-order exit point bounds in exponential last-passage percolation via the coupling technique

Time-time covariance for last passage percolation in half-space

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

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