## ← RandWalk 1.3 TypesofRandomWalks

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Showing Revision 4 created 01/23/2016 by rintu kutum.

1. There are many types of random walks, and
2. I hope... few of them, so you have and idea
3. of the richness.
4. First is the Pearson random walk, in which
5. each step is a fix length, but in a random
6. direction. What I am showing here, is a
7. typical trajectory of such a Pearson random walk.
8. Another example is a Lattice random walk,
9. in which the random walks are constraint
10. to move between nearest neighbour sites of
11. some regular lattice. So, here the steps
12. are fixed length and the direction are
13. either in north, east, south or west.
14. Another type of random walk is so called
15. Levy flight. In the Levy flight, there is a
16. broad distribution of single step lengths
17. but each step is in random direction.
18. Here, we will see that the displacement
19. after many steps can be dominated by the
20. longest single step of the walk.
21. Another example that dear to my heart is
22. the example of Shrinking steps that is a
23. random walker getting lazier and lazier as
24. time is going on and the length of nth-step
25. is landed to end where lambda is less than 1.
26. An amazing aspect of this type of random
27. walk is diversity of type of probability
28. distributions as a function of the Shrinking
29. factor lambda (λ)
30. Know the λ = 0.61, in fact most precisely
31. is the golden ratio, (1 + sqrt(5))/2, the
32. probability distribution is beautiful, self
33. similar pattern that repeats on all scales
34. so the middle blob is same as the entire
35. distribution in inside the middle blob is
36. something which reproduces the entire
37. distribution again
38. Another interesting special case is λ= 0.707 which is actual 1/sqrt(2)
39. Here the probability distribution is
40. made up of 3 linear segments, two tilted
41. lines and one flat line. And there are many
42. other beautiful special cases of this type
43. of random walk Shrinking steps
44. Another important example that appears in
45. nature, turbulent diffusion or random walks
46. that are moving in random convection field
47. In this case, the typical step of length
48. of random walk is a growing with time.
49. And one can get plume like behaviour as you
50. see here from smoke rising from oil fires
51. in the ocean
52. Here are the types of random walks we have
53. just discussed. As we will see, the first 4-types
54. lie in the domain of celebrated central limit theorem
55. in which the probability distribution is
56. asymptotically a Gaussian, independent of
57. details of the microscopic motion. This
58. universality is extremely useful principle
59. in many collective phenomena