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) $|T(x,y) - ET(x,y)|\geq t\sigma_r$, 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 $\mu = \lim_n ET(0,ne_1)/n$, the "nonrandom error" $|\mu|x| - ET(0,x)|$ is at most a constant multiple of $\sigma(|x|)$.