- #1
sha1000
- 124
- 6
- TL;DR
- I have a probability/combinatorics question about random particles on a grid.
Hello,
Take a square lattice with "Ntot" sites (grid). Now place indistinguishable particles "Np" at random. Sampling without replacement.
Np among Ntot is an easy solution
I'm interested in occasions when several particles form a compact “block”. For example, in 2D, 4 particles can form a completely filled square. More generally it can be any squared-cluster of size "s"
My questions:
1) What is the expected (average) number of such fully filled clusters, as a function of "Np", "Ntot" and "s"?
2) And similarly in 3D, what is the expected number of fully filled cubes?
What I tried so far is the following. If occupancies were independent with probability
$$
p = \frac{N_p}{N_{\text{tot}}},
$$
then for an fully occupied square in 2D there are (L-s+1)^2 possible placements, and for an fully occupied cube in 3D there are (L-s+1)^3 possible placements. I tried to work my way through but nothing really works. We'll I also tried to ask GPT, and it gives this solution. But I don't really understand the derivation it proposes, so difficult to judge.
$$
E[C_s] \approx (L-s+1)^2\,p^{s^2} \qquad\text{(2D)},
$$
and
$$
E[C_s] \approx (L-s+1)^3\,p^{s^3} \qquad\text{(3D)}.
$$
Is this remotely correct?
Any help or references would be appreciated.
Thank you.
Take a square lattice with "Ntot" sites (grid). Now place indistinguishable particles "Np" at random. Sampling without replacement.
Np among Ntot is an easy solution
I'm interested in occasions when several particles form a compact “block”. For example, in 2D, 4 particles can form a completely filled square. More generally it can be any squared-cluster of size "s"
My questions:
1) What is the expected (average) number of such fully filled clusters, as a function of "Np", "Ntot" and "s"?
2) And similarly in 3D, what is the expected number of fully filled cubes?
What I tried so far is the following. If occupancies were independent with probability
$$
p = \frac{N_p}{N_{\text{tot}}},
$$
then for an fully occupied square in 2D there are (L-s+1)^2 possible placements, and for an fully occupied cube in 3D there are (L-s+1)^3 possible placements. I tried to work my way through but nothing really works. We'll I also tried to ask GPT, and it gives this solution. But I don't really understand the derivation it proposes, so difficult to judge.
$$
E[C_s] \approx (L-s+1)^2\,p^{s^2} \qquad\text{(2D)},
$$
and
$$
E[C_s] \approx (L-s+1)^3\,p^{s^3} \qquad\text{(3D)}.
$$
Is this remotely correct?
Any help or references would be appreciated.
Thank you.
Last edited: