Construction of a short path in high-dimensional first passage percolation

Olivier Couronné; Nathanaäl Enriquez; Lucas Gerin

doi:10.1214/ECP.v16-1595

Construction of a short path in high-dimensional first passage percolation

Olivier Couronné
, Nathanaäl Enriquez
, Lucas Gerin

Laboratoire MODAL’X 200 avenue de la République
Université Paris-Nanterre

Research output: Contribution to journal › Article › peer-review

8 Citations (Scopus)

Abstract

For first passage percolation in ℤ^d with large d, we construct a path connecting the origin to (x₁ = 1), whose passage time has optimal order log d/d. Besides, an improved lower bound for the “diagonal” speed of the cluster combined with a result by Dhar (1988) shows that the limiting shape in FPP with exponential passage times (and thus that of Eden model) is not the euclidian ball in dimension larger than 35.

Original language	English
Pages (from-to)	22-28
Number of pages	7
Journal	Electronic Communications in Probability
Volume	16
DOIs	https://doi.org/10.1214/ECP.v16-1595
Publication status	Published - 1 Jan 2011
Externally published	Yes

Keywords

First passage percolation
Limit shape
Time constant

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10.1214/ECP.v16-1595

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abstract = "For first passage percolation in ℤd with large d, we construct a path connecting the origin to (x1 = 1), whose passage time has optimal order log d/d. Besides, an improved lower bound for the {\textquotedblleft}diagonal{\textquotedblright} speed of the cluster combined with a result by Dhar (1988) shows that the limiting shape in FPP with exponential passage times (and thus that of Eden model) is not the euclidian ball in dimension larger than 35.",

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AU - Enriquez, Nathanaäl

AU - Gerin, Lucas

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N2 - For first passage percolation in ℤd with large d, we construct a path connecting the origin to (x1 = 1), whose passage time has optimal order log d/d. Besides, an improved lower bound for the “diagonal” speed of the cluster combined with a result by Dhar (1988) shows that the limiting shape in FPP with exponential passage times (and thus that of Eden model) is not the euclidian ball in dimension larger than 35.

AB - For first passage percolation in ℤd with large d, we construct a path connecting the origin to (x1 = 1), whose passage time has optimal order log d/d. Besides, an improved lower bound for the “diagonal” speed of the cluster combined with a result by Dhar (1988) shows that the limiting shape in FPP with exponential passage times (and thus that of Eden model) is not the euclidian ball in dimension larger than 35.

KW - First passage percolation

KW - Limit shape

KW - Time constant

U2 - 10.1214/ECP.v16-1595

DO - 10.1214/ECP.v16-1595

M3 - Article

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VL - 16

SP - 22

EP - 28

JO - Electronic Communications in Probability

JF - Electronic Communications in Probability

ER -

Construction of a short path in high-dimensional first passage percolation

Abstract

Keywords

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