This is not my field, a friend needs the answer for the following question. Suppose we have a decreasing probability function, p: N to [0,1] such that sum_n p(n) = $\infty$. Take the graph where we connect two integers at distance d with probability p(d). Will this graph be connected with probability one?

I see that if the sum is convergent, then we almost surely have an isolated vertex (unless p(1)=1), so this would be "sharp". One possible approach would be to take the path that starts from the origin and if it is at n after some steps, then next goes to the smallest number that is bigger than n and is connected to n and to show that this path has a positive density with probability one. Is this second statement true?

I am sure that these are easy questions for anyone who knows about this.