论文标题
在第一次通过渗透中的均匀波动和徘徊的边界
Uniform fluctuation and wandering bounds in first passage percolation
论文作者
论文摘要
我们考虑在$ \ mathbb {r}^d $中的某些各向同性随机图上的第一次段落渗透。我们假设通过$ | y-x | $是订单$ r $的指数浓度$ t $ t（x，y）$，在某种规模$σ_r$上，$ 0 <χ<χ$ $ r^χ$的订单$ r $。启发式上，这意味着大地测量学的横向徘徊应在$Δ_r=（rσ_r）^{1/2} $的最多按顺序划分。我们表明，实际上，指数浓度和徘徊边界的均匀版本均匀：除了$ t $中的概率很小，没有$ x，y $在自然缸中，$ r $和radius $ r $和radius $kΔ_r$（i）$ | t（x，y） - et（x，y）$ y $ $ $ $ $ $ $ $ $ $ $ $ $ $（II）比从气缸轴的距离$ \ sqrt {t}Δ_r$。我们还确定在时间常数$μ= \ lim_n et（0，ne_1）/n $，“非随机误差” $ |μ| x | -ET（0，x）| $最多是$σ（| x |）$的常数倍数。
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 $σ_r$ whenever $|y-x|$ is of order $r$, with $σ_r$ "growning like $r^χ$" for some $0<χ<1$. Heuristically this means transverse wandering of geodesics should be at most of order $Δ_r = (rσ_r)^{1/2}$. 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 $KΔ_r$ for which either (i) $|T(x,y) - ET(x,y)|\geq tσ_r$, or (ii) the geodesic from $x$ to $y$ wanders more than distance $\sqrt{t}Δ_r$ from the cylinder axis. We also establish that for the time constant $μ= \lim_n ET(0,ne_1)/n$, the "nonrandom error" $|μ|x| - ET(0,x)|$ is at most a constant multiple of $σ(|x|)$.
