Percolation Problem Homework: Probability of Cluster Size s

In summary, the probability that a given site is a member of a cluster of size ##s## in a 1-D lattice with ##L## sites where sites are occupied with probability ##p## is ##s (1-p)^2 p^s##. This is derived by considering the probability that site ##x## is the first member of an ##s##-cluster, which is ##(1-p)^2 p^s##, and multiplying by the number of possible starting sites, which is ##s##.
  • #1
bananabandana
113
5

Homework Statement


We have a 1-D lattice [a line] of ##L## sites. Sites are occupied with probability ##p##. Find the probability that a given site is a member of a cluster of size ##s##. (A cluster is a set of adjacent occupied sites. The cluster size is the number of occupied sites in the cluster)

Homework Equations


[/B]

The Attempt at a Solution


For some site ##x##, I'd say:
$$ Pr( x \in s) = \begin{pmatrix} L \\ 1 \end{pmatrix} \ \frac{ s \times Pr( \text{Make cluster s}) }{L} = sPr(\text{Make Cluster s})$$

##Pr(\text{Make Cluster s})## is the probability that a cluster of size s exists. This is given by (supposing we are not near the ends of the lattice) :
$$Pr(\text{Make Cluster s}) = (1-p)^{2}p^{s}$$

Is this reasoning correct? My textbook also gets to the same answer, but simply states the result, so I am curious [being very rusty on anything to do with statistics] if I have actually done this correctly. (Apologies if this is the wrong forum - I'm aware it's elementary probability theory, but the rest of the text isn't)
 
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  • #2
The probability that a cluster of size s exists in the range depends on L. I guess your Pr(Make Cluster s) is the probability to have a cluster at a given place. In that case the derivation works, as the events are all mutual exclusive you can add the probabilities. This assumes that you are at least s steps away from the border.
 
  • #3
bananabandana said:

Homework Statement


We have a 1-D lattice [a line] of ##L## sites. Sites are occupied with probability ##p##. Find the probability that a given site is a member of a cluster of size ##s##. (A cluster is a set of adjacent occupied sites. The cluster size is the number of occupied sites in the cluster)

Homework Equations


[/B]

The Attempt at a Solution


For some site ##x##, I'd say:
$$ Pr( x \in s) = \begin{pmatrix} L \\ 1 \end{pmatrix} \ \frac{ s \times Pr( \text{Make cluster s}) }{L} = sPr(\text{Make Cluster s})$$

##Pr(\text{Make Cluster s})## is the probability that a cluster of size s exists. This is given by (supposing we are not near the ends of the lattice) :
$$Pr(\text{Make Cluster s}) = (1-p)^{2}p^{s}$$

Is this reasoning correct? My textbook also gets to the same answer, but simply states the result, so I am curious [being very rusty on anything to do with statistics] if I have actually done this correctly. (Apologies if this is the wrong forum - I'm aware it's elementary probability theory, but the rest of the text isn't)

The probability that site ##x## is the first member of an ##s##-cluster is ##(1-p)^2 p^s##, because the site immediately before ##x## must be unoccupied, then sites ##x, x+1, \ldots, x+s## must be occupied and finally site ##x+s+1## must be unoccupied. You get the same probability if site ##x## is the second or third or fourth or ... or last site in the ##s##-cluster. So, indeed, the probability you want is ##s (1-p)^2 p^s##.
 

1. What is the percolation problem?

The percolation problem is a mathematical model used to study the behavior of liquids or gases as they flow through a porous substance. It involves random placement of particles on a grid and determining whether a path can be formed between the top and bottom of the grid.

2. How is probability of cluster size s calculated in the percolation problem?

The probability of cluster size s is calculated by dividing the number of clusters of size s by the total number of clusters in the system. This gives the likelihood of a cluster of size s occurring in the percolation process.

3. What is the significance of cluster size in the percolation problem?

The cluster size in the percolation problem is important because it helps to determine the critical point at which percolation occurs. This critical point is when the cluster size becomes infinitely large, indicating that a path has formed between the top and bottom of the grid.

4. How does the percolation problem relate to real-world situations?

The percolation problem has applications in various fields such as physics, chemistry, and biology. It can be used to model the flow of fluids through porous materials, the spread of diseases, and the behavior of networks and social systems.

5. What are some limitations of the percolation problem?

One limitation of the percolation problem is that it assumes a uniform distribution of particles, which may not be the case in real-world situations. Additionally, it does not take into account the shape or connectivity of the clusters, which can affect the percolation process.

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