The Annals of Probability

First passage percolation has sublinear distance variance

Itai Benjamini,Gil Kalai, and Oded Schramm

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Let $0 < a < b < \infty$, and for each edge e of $\Z^d$ let $\omega_e=a$ or $\omega_e=b$, each with probability $1/2$, independently. This induces a random metric $\dist_\omega$ on the vertices of $\Z^d$, called first passage percolation. We prove that for $d>1$, the distance $\dist_\omega(0,v)$ from the origin to a vertex $v$, $|v|>2$, has variance bounded by $C|v|/\log|v|$, where $C=C(a,b,d)$ is a constant which may only depend on a, b and d. Some related variants are also discussed.

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Ann. Probab. Volume 31, Number 4 (2003), 1970-1978.

First available: 12 November 2003

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 28A35: Measures and integrals in product spaces 60E15: Inequalities; stochastic orderings

Hypercontractive harmonic analysis discrete harmonic analysis discrete cube random metrics discrete isoperimetric inequalities influences


Benjamini, Itai; Kalai, Gil; Schramm, Oded. First passage percolation has sublinear distance variance. The Annals of Probability 31 (2003), no. 4, 1970--1978. doi:10.1214/aop/1068646373.

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