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There are many types of random walks, and

I hope... few of them, so you have and idea

of the richness.

First is the Pearson random walk, in which

each step is a fix length, but in a random

direction. What I am showing here, is a

typical trajectory of such a Pearson random walk.

Another example is a Lattice random walk,

in which the random walks are constraint

to move between nearest neighbour sites of

some regular lattice. So, here the steps

are fixed length and the direction are

either in north, east, south or west.

Another type of random walk is so called

Levy flight. In the Levy flight, there is a

broad distribution of single step lengths

but each step is in random direction.

Here, we will see that the displacement

after many steps can be dominated by the

longest single step of the walk.

Another example that dear to my heart is

the example of Shrinking steps that is a

random walker getting lazier and lazier as

time is going on and the length of nthstep

is landed to end where lambda is less than 1.

An amazing aspect of this type of random

walk is diversity of type of probability

distributions as a function of the Shrinking

factor lambda (λ)

Know the λ = 0.61, in fact most precisely

is the golden ratio, (1 + sqrt(5))/2, the

probability distribution is beautiful, self

similar pattern that repeats on all scales

so the middle blob is same as the entire

distribution in inside the middle blob is

something which reproduces the entire

distribution again

Another interesting special case is λ= 0.707 which is actual 1/sqrt(2)

Here the probability distribution is

made up of 3 linear segments, two tilted

lines and one flat line. And there are many

other beautiful special cases of this type

of random walk Shrinking steps

Another important example that appears in

nature, turbulent diffusion or random walks

that are moving in random convection field

In this case, the typical step of length

of random walk is a growing with time.

And one can get plume like behaviour as you

see here from smoke rising from oil fires

in the ocean

Here are the types of random walks we have

just discussed. As we will see, the first 4types

lie in the domain of celebrated central limit theorem

in which the probability distribution is

asymptotically a Gaussian, independent of

details of the microscopic motion. This

universality is extremely useful principle

in many collective phenomena