Random walks

Random Walks

A random walk is a stochastic process characterized by a sequence of random steps. It's a fundamental concept in various scientific fields.

In a random walk, each step is independent of the previous steps and can follow specific probability distributions.

Historically known as a 'drunkards walk' where a drunk is staggering around in completely random directions.

Real-World Examples of Random Walks

• Physics: Molecule movement in gases and liquids.
• Finance: Stock market fluctuations.
• Biology: Movement patterns of bacteria and other microorganisms.
• Ecology: Animal foraging behavior in nature.

1-D Random Walk

In a 1-dimensional random walk, at each step, the subject moves either one unit forward or one unit backward. This choice is typically made with a 50/50 probability, but can be adjusted for different scenarios.

This simple model is a starting point for understanding more complex random processes.

Let's look at other trajectories:

Continuum behaviour

Individual walkers have a random behaviour but collectively they have some statistical properties.

Similar to the case of the radioactive decay where each atom is independent.

100 walkers with 100 steps:

1000 walkers with 100 steps:

To quantify if let's look at the average position and the standard deviation as a function of the number of steps:

$$\bar x = \left<x\right> \;, \qquad \left<(x-\bar x)^2\right>$$

average position

The average position is zero. Let's look at the squared distance:

Random walks in multiple dimensions

We can have random walks in more than one dimension

• at each step we decide
  • in which dimension to move
  • whether to move in the positive or negative direction

3D walkers

Squared distance in multiple dimensions

The scaling of the squared distance as a function of the number of steps is the same in all dimensions:

Final points

Reflections.

I massively underestimated the amount of admin involved on both the front and back end.

I've anecdotally heard people spending 4-5 hours a week on these assignments!

This is a huge amount but at least you do it; very easy to kid yourself that you know things

Please consider clicking the link at the top of the assignment.

35 years old!

It is later than you think!

Thank you!

Every minute after 6 pm was voluntary from the PGR demonstrators!

Apologies if I've upset you; I have no authority to hand out concessions regardless - I set a bad precedent at the start - deadlines are final

The first (and hopefully(#!?)) last lecture course I've done

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