Sublinear variance in first-passage percolation for general distributions

By Michael Damron, Jack Hanson, Philippe Sosoe. 2013

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We prove that the variance of the passage time from the origin to a point $$x$$x in first-passage percolation on $$\mathbb {Z}^d$$Zd is sublinear in the distance to $$x$$x when $$d\ge 2$$d≥2, obeying the bound $$C\Vert x\Vert /\log \Vert x\Vert $$C‖x‖/log‖x‖, under minimal assumptions on the edge-weight distribution. The proof applies equally to absolutely continuous, discrete and singular continuous distributions and mixtures thereof, and requires only $$2+ \log $$2+log moments. The main result extends work of Benjamini–Kalai–Schramm (Ann Prob 31, 2003) and Benaim–Rossignol (Ann Inst Henri Poincaré Prob Stat 3, 2008).

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/266ea3ed820b862ae2069a8e58b318a1ded5f59a