Let me now turn to a foundational result in the theory of random walks
known as the central limit theorem.
So let's imagine thinking about a random walk in arbitrary spatial dimensions
in which the single step is given by a displacement, little x, and the
probability distribution of these single
step displacements is given by little p of little x.
Let us assume that these little x's are iid variables. This is a standard acronym
for independent, identically distributed.
So, independent and identically distributed.
So that is, for each x, p of x is the same distribution function.
Let us further assume that the mean displacement in a single step,
so we no longer have to restrict ourselves to a symmetrical random walk.
Let me assume that the mean displacement in a single step is finite,
very mild restriction.
And let me also assume that the mean square displacement in a single step
is also finite.
Under these conditions, then the probability distribution after n steps,
so P of n after n steps, of the total displacement.
So here, big X is the sum of the displacements, little x, n of these guys.
Then, this probability distribution of the total displacement is equal to
one over square root of 2 pi n sigma squared, e to the minus x, minus n times
the average of little x squared, divided by 2, 2 n sigma squared,
where here sigma squared is equal to the variance in the single step distribution.
So, it's the difference between x squared average minus x average squared.
So, this statement is known as the central limit theorem.
And I should say that it's holding and the limit is n is going to infinity.
This is a very powerful and general result because one needs then only the first two
moments of the single step distribution, these are the only things that matter to
determine the long distance, long-time properties of a random walk.
So, this is an example of a universal statement in which short-range details of
the microscopic process, namely the details of the single steps basically do
not matter in the long-time limit.
It doesn't matter if we are talking about a nearest neighbor random walk in one
dimension or some other type of hopping distribution as long as the distribution
obeys these two conditions, that the first and second moments are finite, then the
distribution of the long-time limit is this universal Gaussian form.
So, notice also that from this result, we now know that the mean displacement
after n steps of a random walk, which is just the integral of x p n of x, d x is
equal to n times the displacement after a single step.
And similarly, the variance after many steps of the random walk, x squared
average minus x average squared.
Again using the same definition as before, so this is the integral of x squared
p n of x, d x minus x average squared, which we've already computed.
This thing is simply equal to n sigma squared. So, once we know the first and
second moments of the single step distribution, what they are, then we know
that the distribution of a random walk of n steps is given by a Gaussian form,
and from there we can compute any moment we want of the random walk itself.