On the truncated long-range percolation on $\Z^2$

We consider an independent long-range bond percolation on Z2. Horizontal and vertical bonds of length n are independently open with probability p_n ¿ [0, 1]. Given ¿n=18¿i=1n(1 - pi) <8, we prove that there exists an infinite cluster of open bonds of length less than or equal to N for some large but finite N. The result gives a partial answer to the truncation problem.