Why 8 probabilities in 3D random walk?

[jaobyccdee]

Why are there 8 possible moves in 3D random walk?
With R being the distance from the origin, how is its relationship with the number of steps and dimensionality? Thx!

 

[chiro]

Why are there 8 possible moves in 3D random walk?
With R being the distance from the origin, how is its relationship with the number of steps and dimensionality? Thx!

Hey jaobyccdee and welcome to the forums.

The easiest way is to think in terms of each dimension. In a one-dimensional random walk, we have two probabilities: probability of going up, and the probability of giong down.

Now consider that each dimension is independent from the other. For three dimensions we have 3 sets of probabilities which means 2+2+2 different probabilities if we allow arbitrary values.

However you should be aware that we only have to specify three values since we can get the other three by using the 1 - prob rule.

The easiest way to see why its 2^3 = 8 different combinations is by considering three bits. Each bit is independent of each other and allows two different values: up or down for each dimension.

If you count the number of possible values you get 8. If you are not quite sure, draw a tree diagram of all the possible states up and down for every dimension and you should end up with eight possible states. One example state is {up,up,down} which corresponds to an up value for dimension 1, an up value for dimension 2 and a down value for dimension 3.

 

[jaobyccdee]

thx for replying. But i still have a question. What's the difference then between having three steps in 1D and 1 step in 3D?

 

[chiro]

thx for replying. But i still have a question. What's the difference then between having three steps in 1D and 1 step in 3D?

A random walk by definition talks about probabilities of going up and down (or positive and negative): We are talking about deltas, or rather changes from an existing value at some time, to the next time increment.

By the nature of a random walk, you can't have three steps in one dimension: the only way for something to go is up or down.

Also you have to remember that every dimension is completely independent from the others. The x-dimension is completely independent from the y-dimension and the z-dimension.

The easiest way to describe this is by considering a point in three-dimensional space.

Lets say you have a point <x,y,z>. Now if you change the x-coordinate from x to x0, your new point is <x0,y,z>. Note that by changing the x to x0, the y and the z haven't changed: in other words they are independent.

If you wanted to have say all them connected, then all of them would have to be a function of one variable.

As an example let's consider our point example again: Let's say we have a point that is determined by <t,t + 2, t + 3>. If we change t to t0, that ends up affecting every co-ordinate by changing the value to <t0,t0 + 2, t0 + 3> which shows that this is not independent.

Basically if you had this kind of dependency for a random walk in three dimensions, you would have one random variable, call it T and three random variables that are functions of T.

If we use our analog example of the point, our "dependent random walk" (its not a real random walk, but I use the analogy to help our discussion) would have the random variables x(T), y(T), z(T) where x,y,z are functions of our random variable T. Note that these are all dependent on T.

But since every component in a 3D random walk is independent, that means we have three random variables instead of 1. So instead of our x(T), y(T), z(T) scenario we have x(A), y(B), z(C) where A,B,C are all independent. This is why we have a lot more probabilities in a 3D random-walk (2x2x2 as opposed to just 2).

 

[jaobyccdee]

Thx a lot! So, if we have {up, up, down}, does it mean we have only walked one step and should see it as a vector (up, up, down} in 3D space? So, R (the position) does depend on the dimensionality? But then how does the position relate to the number of steps? I think i am confused by the number of steps and number of dimension, like the difference in their calculations.

 

[SW VandeCarr]

Thx a lot! So, if we have {up, up, down}, does it mean we have only walked one step and should see it as a vector (up, up, down} in 3D space? So, R (the position) does depend on the dimensionality? But then how does the position relate to the number of steps? I think i am confused by the number of steps and number of dimension, like the difference in their calculations.

Do you know how to add vectors? Given an origin and 3 dimensions, every point in the space is identified with a coordinate tuple (x,y,z) Consider the starting position to be defined as (0,0,0). As the particle moves from one position to another, its position is described by a unique tuple. In the Euclidean case, you can consider this a "jump" without regard to its motion between positions. Each jump is a step. Also, the value of the tuple at any point can tell you how far the particle has traveled from the origin. The path of the particle can be reconstructed from the history of its prior locations, represented by vectors placed "head to tail" where the tail is the previous location and the head is the new location at each step. The net distance traveled from the origin (or any point in the path) can be calculated using the Pythagorean Theorem and the information from the relevant tuples.

If this was 2 dimensional, the tuple would have two values (x,y), one dimension, one value (x).