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klotza submitted a new PF Insights post
Fun with Self-Avoiding Walks
Continue reading the Original PF Insights Post.
Fun with Self-Avoiding Walks
Continue reading the Original PF Insights Post.
"The size of a random walk, being random, has a Gaussian distribution" I would think it would be a Poisson distribution. In this circumstance it would approximate a Gaussian though.klotza said:klotza submitted a new PF Insights post
Fun with Self-Avoiding Walks
Continue reading the Original PF Insights Post.![]()
klotza said:Hi Jimster41 I just saw your questions now, several months after you asked them.
The self-avoiding walk only grows from the end, it has linear topology in its basic form. Other versions are bottle-brush polymers or comb polymers, hyperbranched polymers, dendritics, etc, which have branchings.
I don't know much about self organizing criticality and scale-invariance, but I do know that regular brownian diffusion can be described by a fractal wiener process. The mathematical literature on the critical behaviour of SAWs is quite extensive but a bit beyond me.
mfb said:A walk ends if there is no free spot any more?
Up to symmetries, there is just one 7-step walk (up, left, left, down, down, right, up), and two 8-step walks (same pattern starting one step later). The 8-step walk needs a more precise arrangement of the steps.
mfb - Again many thanks for taking an interest in this problem. The wheels in my 82 year old head don't spin as fast as they once did! Your values for the probabilities, P(7) and P(8), of 7 and 8 step walks agree well with two 1-million walk simulation runs;mfb said:I see an effect from parity (like black/white checkerboard fields), but I would not expect it to be so large: a walk after an odd number of steps N can potentially be blocked by (N+1)/2 other elements, while a walk after an even number of steps can be blocked by N/2 other elements.
The probability of a walk ending after 7 steps is 6/37, while the probability of ending after 8 steps is 5/37.
The probabilities are absolute. If we would have those values for all N, they would add to 1.herb helbig said:I suppose, more accurately, the 8-step probability should be reduced by multiplying by the probability of not being trapped at 7 steps: 1-P(7) = 0.99726, but it would take a simulation of about 50 million walks to verify that, and much confidence in the random number generator!
That (n+1)/2 vs n/2 pattern should have an odd/even pattern.herb helbig said:The trend of your (n+1)/2, N/2 parity effect is evident in the data, but apparently an underestimate.
I have a figure summarizing what I have said, but not sure whether it has uploaded.
A self-avoiding walk is a mathematical concept that describes a path or trajectory taken by a point or particle that does not cross or intersect with any of its previous points. This is often modeled on a grid or lattice and is used in various fields such as physics, chemistry, and computer science.
Self-avoiding walks have been extensively studied in various scientific fields as they provide a simple yet powerful way to understand the behavior of complex systems. They have been used to model polymer chains, protein folding, and the spread of diseases, among other things. They also have practical applications in computer algorithms and simulations.
Yes, self-avoiding walks have been successfully applied in various real-world scenarios, such as traffic flow, molecular dynamics, and social networks. They have also been used to study the behavior of physical systems, such as fluids and gases.
While self-avoiding walks have been proven to be useful in many applications, they also have some limitations. One of the main limitations is that they are often simplified versions of real-world systems and may not accurately represent all aspects of the system. They also require a considerable amount of computing power and can be computationally expensive for large systems.
Self-avoiding walks are an active area of research, and scientists are constantly seeking to improve upon and develop new methods for using them. Some current research topics include incorporating different types of constraints, such as energy minimization, into the model and studying self-avoiding walks on non-Euclidean geometries. Additionally, advancements in computing power and algorithms are allowing for the study of larger and more complex systems.