论文标题
随机假计和应用
Random pseudometrics and applications
论文作者
论文摘要
令$ t $为$ \ mathbb r^d $上的随机ergodic伪计。此设置概括了经典\ emph {第一个段落渗透}（fpp）$ \ mathbb z^d $。我们提供了$ t $的简单条件，即即时单臂和指数级独立性的衰减，以确保其时间常数的积极性，这几乎可以肯定的是，$ t $从起源中给出的伪距离是渐近的。将此一般结果与以前已知的结果相结合，我们证明 在与高斯FPP（包括天然的Bargmann-Fock模型）的呈正相关的磁场的情况下，高斯渗透的已知相转换是与指数快速的衰减相关性的； Voronoi渗透的已知相跃迁也扩展到相关的FPP。 对于带指数尾巴的半径的布尔渗透也是如此，这一结果是没有这种情况的。 我们证明了对随机连续riemannian指标的常数的积极性，包括在维度上具有无限相关的案例$ d = 2 $。 最后，我们表明，如果存在的单臂的关键指数在上面以$ d-1 $限制。这可以容纳Bernoulli渗透，平面高斯田，平面Voronoi渗透以及带指数小尾巴的布尔渗透。
Let $T$ be a random ergodic pseudometric over $\mathbb R^d$. This setting generalizes the classical \emph{first passage percolation} (FPP) over $\mathbb Z^d$. We provide simple conditions on $T$, the decay of instant one-arms and exponential quasi-independence, that ensure the positivity of its time constants, that is almost surely, the pseudo-distance given by $T$ from the origin is asymptotically a norm. Combining this general result with previously known ones, we prove that The known phase transition for Gaussian percolation in the case of fields with positive correlations with exponentially fast decayholds for Gaussian FPP, including the natural Bargmann-Fock model; The known phase transition for Voronoi percolation also extends to the associated FPP; The same happens for Boolean percolation for radii with exponential tails, a result which was known without this condition. We prove the positivity of the constant for random continuous Riemannian metrics, including cases with infinite correlations in dimension $d=2$. Finally, we show that the critical exponent for the one-arm, if exists, is bounded above by $d-1$. This holds forbond Bernoulli percolation, planar Gaussian fields, planar Voronoi percolation, and Boolean percolation with exponential small tails.
