arXiv 1706.025972017-06math.PR

无标度渗流中的爆炸和距离

Explosion and distances in scale-free percolation

Remco van der Hofstad, Julia Komjathy

备注：: 40 pages. We fixed some typos, and shortened the paper

摘要：

研究了$\mathbb Z^d$上的加权无标度渗流模型。在SFPW模型中，$\mathbb Z^d$的顶点被指定为i.i.d.。加权$(W_x)_{x\in \mathbb Z^d}$，遵循具有尾部指数$τ>1$的幂律分布。以权重集合为条件，边$(x,y)_{x, y \in \mathbb Z^d}$独立地存在，对于某些$α, λ>0$，具有参数$λW_x W_y/(\|x-y\|)^α$的泊松随机变量至少为1的概率，并且其中$\|\cdot \|$表示欧几里得距离。在以这种方式构造图之后，我们分配独立同分布（i.i.d.）从分布$L$到图中所有现有边的随机边长度。本文的重点是确定什么时候得到的模型是\emph{explosive}，即什么时候有可能从一个顶点在有限时间内到达无穷多个顶点。我们表明，爆炸恰好发生在那些边长分布，产生爆炸性的分支过程与无限平均幂律后代分布。对于非爆炸边长分布，当$γ\in (1,2)$时，我们证明了到达距离$n$的第一个顶点所需时间的渐近行为。对于$γ>2$，我们证明了$t$从原点到时间的可达顶点数至多呈指数增长，因此爆炸是永远不可能的。对于非爆炸边长分布，当$γ\in (1,2)$时，我们进一步确定了距离的一阶渐近性，当$γ\in (1,2)$。作为一个推论，我们得到了一个尖锐的上限和下限的图距离，关闭之间的差距在Deijfen霍夫斯塔德'13时，$γ\in(1,2), τ>2$的图距离的下限和上限。

英文摘要：

We investigate the weighted scale-free percolation (SFPW) model on $\mathbb Z^d$. In the SFPW model, the vertices of $\mathbb Z^d$ are assigned i.i.d. weights $(W_x)_{x\in \mathbb Z^d}$, following a power-law distribution with tail exponent $τ>1$. Conditioned on the collection of weights, the edges $(x,y)_{x, y \in \mathbb Z^d}$ are present independently with probability that a Poisson random variable with parameter $λW_x W_y/(\|x-y\|)^α$ is at least one, for some $α, λ>0$, and where $\|\cdot \|$ denotes the Euclidean distance. After the graph is constructed this way, we assign independent and identically distributed (i.i.d.) random edge lengths from distribution $L$ to all existing edges in the graph. The focus of the paper is to determine when is the obtained model \emph{explosive}, that is, when it is possible to reach infinitely many vertices in finite time from a vertex. We show that explosion happens precisely for those edge length distributions that produce explosive branching processes with infinite mean power law offspring distributions. For non-explosive edge-length distributions, when $γ\in (1,2)$, we characterise the asymptotic behaviour of the time it takes to reach the first vertex that is graph distance $n$ away. For $γ>2$, we show that the number of vertices reachable by time $t$ from the origin grows at most exponentially, thus explosion is never possible. For the non-explosive edge-length distributions, when $γ\in (1,2)$, we further determine the first order asymptotics of distances when $γ\in (1,2)$. As a corollary we obtain a sharp upper and lower bound for graph distances, closing a gap between a lower and upper bound on graph distances in Deijfen Hofstad `13 when $γ\in(1,2), τ>2$.

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摘要：

英文摘要：

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