Uniform fluctuation and wandering bounds in first passage percolation

Abstract

We consider first passage percolation on certain isotropic random graphs in ${\mathbb{R}}^d$. We assume exponential concentration of passage times $T(x,y)$, on some scale ${\sigma }_r$ whenever $|y-x|$ is of order $r$, with ${\sigma }_r$ "growning like $r^{\chi }$" for some $0<\chi <1$. Heuristically this means transverse wandering of geodesics should be at most of order ${\Delta }_r=(r{\sigma }_r{)}^{1/2}$. We show that in fact uniform versions of exponential concentration and wandering bounds hold: except with probability exponentially small in $t$, there are no $x,y$ in a natural cylinder of length $r$ and radius $K{\Delta }_r$ for which either (i) $|T(x,y)-ET(x,y)|\ge t{\sigma }_r$, or (ii) the geodesic from $x$ to $y$ wanders more than distance $\sqrt{t}{\Delta }_r$ from the cylinder axis. We also establish that for the time constant $\mu ={\lim }_{n}ET(0,n{e}_1)/n$, the "nonrandom error" $|\mu |x|-ET(0,x)|$ is at most a constant multiple of $\sigma (|x|)$.

arXiv comment89 pages, 12 figures. Lemma 1.2 added. Misc small corrections and clarifications

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