Understanding Percolation Threshold p_c on Triangular Lattices

[bubbloy]

i am having some trouble understanding the meaning of what a percolation threshold is p_{c}.
apparently on triangular lattices a threshold of 0.5 is the result on any sized lattice.
however i can definitely think of a way to fill in half the points on a triangle lattice and not have it span across the lattice.
similarly, one could make a zig zag line of connected sites to span a lattice without using anywhere near 1/2 the points. so is the percolation threshold a number at which you can expect to see percolation?

all the books I am reading seem to say that above it everything has a percolating net and below it there are no possible percolating nets.

thanks a lot,

josh S

 

[bubbloy]

well in case anyone was wondering, the percolation threshold is the probability with which you need to fill in the nodes on a lattice so that in the limit of an infinite lattice, there is a non zero chance that you will have a spanning cluster going across it. for example, one could construct a spanning cluster across an infinite lattice by connecting a line of edges across the whole thing and in this example the probability you fill the lattice with is 0 (one row/infinite rows) but the probability of this happening is so low that it does not survive in the infinite limit.