Random Walk in confined region and loop configurations

In summary: So if you have a line at x=0, you can reflect your entire path through that line and count how many times it touches the line. In the 1-D case, that's easy. In the 2-D case, this principle will help you, but you'll have to think a little harder about it.In summary, the conversation discusses a random walk on a 2-dimensional square lattice with a finite size. The walker cannot cross the boundary and has a fixed step length. The question is whether there has been any work done on this type of random walk and how many ways there are for the walker to be at the starting position after a fixed number of steps. The conversation also mentions the concept of self-
  • #1
Suppose I take a random walk on a 2 dimensional square lattice, but this lattice plane has a finite size, e.g. Dx*Dy. I can not cross the boundary, my step length is the lattice cell size, I either go straight or make turns with right angle. Is there any work on this type of random walk?

If the total walking distance is fixed, e.g. 2N, in this confined region, how may ways are there that at the 2Nth step I am at the starting position? Or equivalently, how many configurations of loops of length 2N are there in this confined square lattice? How about if this loop is a self avoiding loop?

Anybody knows related work on this? Since my major is physics, I am not sure if some mathematicians have done this. Thanks in advance.
 
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  • #2
welcome to pf!

hi allanqunzi! welcome to pf! :smile:
allanqunzi said:
If the total walking distance is fixed, e.g. 2N, in this confined region, how may ways are there that at the 2Nth step I am at the starting position?

if it was a 1D lattice, the answer would be easy …

just write 1 for left and 0 for right, and count the number of ways of writing N 1s and N 0s :wink:

you should be able to find a similar method for 2D :smile:

(alternatively, googling "random walk" and "lattice" should give you plenty of hints)
 
  • #3
Definitely try it for the 1-D case first if you haven't already. Even that requires some thought. If there were no boundaries, both the 1-D and 2-D cases would be easy. But you have to subtract those paths that cross a boundary. To find those, use the method of images (or principle of reflection). If you have two points, A and B, on the same side of a line, the number of paths between those two points that cross or touch the line is the same as the total number of paths between A* and B, where A* is the reflection of A through that line.
 

1. How does the random walk process work in a confined region?

In a random walk, an object moves in a random direction with equal probability at each step. In a confined region, the object is restricted to stay within a certain boundary, such as a box or a circle. The object will continue to move in a random direction until it reaches the boundary, at which point it must change direction to remain within the confined region.

2. What is the purpose of studying random walks in confined regions?

Studying random walks in confined regions can provide insight into various physical phenomena such as diffusion, polymer dynamics, and reaction rates. It can also be used to model real-world scenarios, such as the movement of particles in a cell or the spread of a disease in a population.

3. How are loop configurations incorporated into the random walk process?

Loop configurations refer to the shape or arrangement of the confined region. In a random walk, the loop configuration can affect the probability of the object reaching certain areas within the confined region. This can be taken into account by adjusting the probabilities of the random walk in different directions.

4. Can the random walk process be used to model complex systems?

Yes, the random walk process can be used to model complex systems by incorporating additional factors such as interactions between objects, external forces, and varying probabilities for different types of movement. This allows for a more realistic representation of the system and can provide valuable insights and predictions.

5. How is the behavior of random walks in confined regions affected by different parameters?

The behavior of random walks in confined regions can be affected by parameters such as the size and shape of the confined region, the step size of the object, and the probability of changing direction. Varying these parameters can result in different patterns and outcomes of the random walk process.

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