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.

 

[charng]

The percolation threshold, denoted as p_c, is a critical value that represents the point at which a system transitions from a disconnected state to a connected state. In the context of triangular lattices, it is the probability at which a randomly chosen lattice site is occupied, leading to the formation of a percolating cluster that spans the entire lattice.

The value of p_c is dependent on the specific lattice structure and can vary for different types of lattices. In the case of triangular lattices, p_c is known to be 0.5, meaning that if half of the lattice sites are occupied, there is a high likelihood of a percolating cluster spanning the lattice.

It is important to note that p_c is a statistical measure and does not guarantee the formation of a percolating cluster in every single instance. As you mentioned, there are ways to fill in half of the lattice points without creating a percolating cluster. However, on average, at p_c, there is a high chance of a percolating cluster forming.

In summary, the percolation threshold is a critical value that represents the point at which a system transitions from a disconnected state to a connected state. In the case of triangular lattices, it is a statistical measure and not a guarantee of percolation in every instance. It is used as a reference point to understand the behavior of percolation in different lattice structures.