Year: 2001
Type: preprint
Abstract: We consider the following long range percolation model: an undirected graph with the node set $\{0,1,...,N\}^d$, has edges $(\x,\y)$ selected with probability $\approx β/||\x-\y||^s$ if $||\x-\y||>1$, and with probability 1 if $||\x-\y||=1$, for some parameters $β,s>0$. This model was introduced by Benjamini and Berger, who obtained bounds on the diameter of this graph for the one-dimensional case $d=1$ and for various values of $s$, but left cases $s=1,2$ open. We show that, with high probability, the diameter of this graph is $Θ(\log N/\log\log N)$ when $s=d$, and, for some constants $02d$. We also provide a simple proof that the diameter is at most $\log^{O(1)}N$ with high probability, when $d
Source: arXiv (Cornell University)
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4394670962