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
There are many types of random walks, and
are fixed length and the direction are
Another example is a Lattice random walk,
in which the random walks are constraint
to move between nearest neighbour sites of
typical trajectory of such a Pearson random walk.
some regular lattice. So, here the steps
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 nth-step
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 4-types
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