Uniform fluctuation and wandering bounds in first passage percolation

We consider first passage percolation on certain isotropic random graphs in 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)| >= t sigma (r), or (ii) the geodesic from x to y wanders more than distance root t Delta(r) from the cylinder axis. We also establish that for the time constant mu = limn ET (0, ne(1))/n, the "nonrandom error" ET (0, x) - mu |x| is at most a constant multiple of sigma (|x|).