Subdiffusive concentration in first-passage percolation

Abstract

We prove exponential concentration in i.i.d. first-passage percolation in $Z^d$ for all $d\ge 2$ and general edge-weights $(t_e)$. Precisely, under an exponential moment assumption $E{e}^{\alpha t_e}<\infty$ for some $\alpha >0$) on the edge-weight distribution, we prove the inequality $$P(|T(0,x)-ET(0,x)|\ge \lambda \sqrt{\frac{|x|}{log|x|}})\le c{e}^{-c^{\prime }\lambda },|x|>1$$ for the point-to-point passage time $T(0,x)$. Under a weaker assumption $E{t}_e^2(\log t_e{)}_{+}<\infty$ we show a corresponding inequality for the lower-tail of the distribution of $T(0,x)$. These results extend work of Benaim-Rossignol to general distributions.

31 pages, the main discrete derivative bound is now simplified, formulated in terms of integrating over uniform variables

References in corpus (4)

• First passage percolation and competition models
• Percolation since Saint-Flour
• Subgaussian concentration and rates of convergence in directed polymers
• Rate of convergence in first-passage percolation under low moments

Cited by in corpus (5)

• Self-organized Segregation on the Grid
• Uniform fluctuation and wandering bounds in first passage percolation
• Localization and free energy asymptotics in disordered statistical mechanics and random growth models
• Sublinear variance in Euclidean first-passage percolation
• Power Lower Bounds for the Central Moments of the Last Passage Time for Directed Percolation in a Thin Rectangle