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.