A striking feature of random walks
is that the spatial dimension , little d,
plays a crucial role.
We normally think of
random walks as moving about in one
dimension, two dimensions, or three
dimensions, but it behooves us to extend
ourselves and think about a random walk
in arbitrary spatial dimension d,
and as we'll see as a function of d there
is a transition in a fundamental
property of a random walk.
So let's think of a typical Pearson random
walk and now let me draw in a continuum
formulation, so it's some continuous
trajectory that that starts somewhere
and moves about in a random walk fashion,
and let it go until time t. And
as we've learned, the RMS displacement
scales at the square root of t,
so there's a characteristic distance
that grows at the square root of time.
So, we can define an exploration sphere
as the range over which a typical
random walk can be expected
to extend up to time t.
Lets now calculate the density of points
visited inside this exploration sphere.
So the density
of visited sites
So let me call that quantity Ρ.
So what is this density of visited sites?
It's the number of visited sites, divided
by the volume of the exploration sphere.
So in a time t,
our random walker will
visit t sites,
and then we have to divide
by the volume of the exploration sphere
which is its radius, which is square root
of t to the power, dimension of space,
times some...
...constant numerical factors
that're irrelevant for this consideration.
the crucial point is that this quantity
has a interesting dependence
on time. It is t to the power of 1-d/2
So this density of visited sites has three
fundamental behaviors that depend on the
spacial dimension. So if I look in summary
at how P depends on time, there are three
answers.
For d > 2 this density will go to
0, because for d > 2 this is a negative
exponent so in the infinite time limit,
this density goes to 0. So I should maybe
write → here for approaching in limit of t
going to infinity. So this happens for
d > 2. For d < 2 on the other hand, then
this is a positive exponent and so this
density of visited sites is approaching
inifinity.
In d=2 this exponent is equal to 0
and so it would suggest that the density
approach is a constant for d = 2.
This change in behavior between infinite
density and zero density is also what's
known as a transition between transience
and recurrence.
So for d < 2, this regime
is what's called "recurrent behavior".
And what is meant by recurrent behavior
is that because the density of points is
infinite, it means a random walker as
it's moving about will visit every single
site. So in particular, if you start at
some site, you are guaranteed that you
must return eventually to that site.
So the recurrent regime is also the regime
where the return of the random walker
is certain.
Conversely, for d > 2, this is
what's called the "transient regime".
and in the transient regime, if you launch
a random walker from a given point,
because the density of points is going to
0, it's not certain that it's going to
return to its starting point.
So here, return is uncertain.
The transition at d = 2 is actually lying
in the recurrent regime, because it turns
out that this constant behavior
is the result
of this very crude,
is a flaw of this
very crude line of reasoning, and
it turns out that the correct behavior of
the density is that it goes like log t in
two dimensions.
So the fundamental result of this
discussion here is that for d ≤ 2,
a random walk is recurrent
and return is certain.
For d > 2, a random walk is transient,
and its return is uncertain.
One important point to emphasize is that
even though the return is certain for
d ≤ 2, as we will learn at the end of this
tutorial, the mean time to return to the
origin is actually infinite.
This dichotomy between certain return and
infinite return time is what makes random
walks so intriguing.