Rate of convergence in first-passage percolation under low moments

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• 作者：Michael Damron^a ; ^{mdamron6@gatech.edu" class="auth_mail" title="E-mail the corresponding author} ; Naoki Kubota^b
• 关键词：First-passage percolation ; Concentration inequalities ; Low moments ; Rate of convergence
• 刊名：Stochastic Processes and their Applications
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：126
• 期：10
• 页码：3065-3076
• 全文大小：260 K

文摘

We consider first-passage percolation on the d$d$ dimensional cubic lattice for d≥2$d\ge 2$; that is, we assign independently to each edge e$e$ a nonnegative random weight t_e$t_e$ with a common distribution and consider the induced random graph distance (the passage time), T(x,y)$T(x,y)$. It is known that for each x∈Z^d$x\in {\mathbb{Z}}^d$, μ(x)=lim_nT(0,nx)/n$\mu \left(x\right)={lim}_{n}T(0,nx)/n$ exists and that $0\le \mathbb{E}T(0,x)-\mu \left(x\right)\le C{\Vert x\Vert }_1^{1/2}log{\Vert x\Vert }_1$ under the condition Ee^αt_e<∞$\mathbb{E}{e}^{\alpha t_e}<\infty$ for some α>0$\alpha >0$. By combining tools from concentration of measure with Alexander’s methods, we show how such bounds can be extended to t_e$t_e$’s with distributions that have only low moments. For such edge-weights, we obtain an improved bound C(‖x‖₁log‖x‖₁)^1/2$C{({\Vert x\Vert }_{1}log{\Vert x\Vert }_1)}^{1/2}$ and bounds on the rate of convergence to the limit shape.