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First passage percolation has sublinear distance variance. (English) Zbl 1087.60070
Summary: Let $$0 < a < b < \infty$$, and for each edge e of $$\mathbb Z^d$$ let $$\omega_e=a$$ or $$\omega_e=b$$, each with probability $$1/2$$, independently. This induces a random metric $$\text{dist}_\omega$$ on the vertices of $$\mathbb Z^d$$, called first passage percolation. We prove that for $$d>1$$, the distance $$\text{dist}_\omega(0,v)$$ 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.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 28A35 Measures and integrals in product spaces 60B15 Probability measures on groups or semigroups, Fourier transforms, factorization 60E15 Inequalities; stochastic orderings
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