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 $\mathit{\sigma }(r)$ whenever $|y-x|$ is of order r, with $\mathit{\sigma }(r)$ “growning like $r^{\mathit{\chi }}$” for some $0<\mathit{\chi }<1$. Heuristically this means transverse wandering of geodesics should be at most of order $\mathrm{\Delta }(r)={(r\mathit{\sigma }(r))}^{1∕2}$. 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 $K\mathrm{\Delta }(r)$ for which either (i) $|T(x,y)-ET(x,y)|\ge t\mathit{\sigma }(r)$, or (ii) the geodesic from x to y wanders more than distance $\sqrt{t}\mathrm{\Delta }(r)$ from the cylinder axis. We also establish that for the time constant $\mathrm{\mu }={lim}_{n}ET(0,n{e}_1)∕n$, the “nonrandom error” $ET(0,x)-\mathrm{\mu }|x|$ is at most a constant multiple of $\mathit{\sigma }(|x|)$.