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