*A shorter version of this post was published on The Conversation on 19th December 2013.*

“So much of life, it seems to me, is determined by pure randomness” – Sidney Poitier

The warmth of sunlight on your face, the view from your backyard – such delights are delivered to you by countless pieces of sunlight that you constantly absorb. Each such photon of sunlight has come to the end of an incredible journey. Its journey began with a violent birth in the nuclear fires of the sun’s core, which was followed by a long migration to the solar surface and a final leap across the 150 million kilometre void of space to reach the Earth (and you).

Still more remarkable is the time required for such a journey. Travelling at the fastest speed known to physics, the photon crossed from the sun to the Earth in a mere 9 minutes. In contrast, the first 0.05% of the journey – just from the sun’s core up to its surface – lasted almost *10 thousand million* times as long, taking an average of 170 000 years to complete.

## Discrete random walks

“Creativity is the ability to introduce order into the randomness of nature” – Eric Hoffer

*discrete one-dimensional random walk*, in which the walker only moves back and forth along one particular direction (and takes the same sized step every time). For instance, we might position ourselves at a starting point (the origin) and take one hundred steps, using a coin flip at each step to determine our next move. If the coin comes up heads, we’ll move one metre to the north; if tails, one metre south. If we keep track of our progress and repeat the exercise eight different times, we might end up with the eight random walks shown below:

*central limit theorem)*tells us that for walks like these, the probability of being at a particular spot after a certain number of steps closely follows the normal distribution. In fact, if we look at the 100th row of Pascal’s Triangle we see an orderly row of 100 different numbers, each of which corresponds to one of the locations we could have ended our walk on, and each of which tells us how many different random walks (of 100 steps) we could have taken to end up at that particular spot. So studying Pascal’s triangle shows us exactly how likely we are to end up at a particular location – the bigger the number in that spot, the more likely it is that your random walk will lead you there.

*level-crossing phenomena*– if allowed to go on forever, a simple random walk like this will cross every point (including the origin) infinitely many times. So yes, we are guaranteed to return to our starting point –

*eventually*. The level-crossing phenomenon is also known as the gambler’s ruin, since if our random walk represents the bank balance of a gambler who is playing against a house with unlimited funds, the gambler’s balance is guaranteed to cross zero eventually – after which the game has ended, and the gambler has lost.

## Non-discreet drunken walks

“A drunk man will find his way home, but a drunk bird may get lost forever.” – Shizuo Kakutani

*discrete two-dimensional random walk*(the steps are still the same size, but now the walker can move randomly on a two-dimensional grid). Not only do they capture the wanderings of over-indulgent partygoers, random walks such as these (and their higher-dimensional counterparts) are the basis on which nearly all random activity is modelled – from the wanderings of foraging animals to the twists and turns of chemical polymers.

*Nature*asking for assistance with determining the probability of finding a random walker at a specific place after a certain number of steps. Pearson went on to use random walks to develop mathematical models of mosquito infestation. His letter was answered by the Nobel-winning physicist Lord Rayleigh, who had already analysed a more general version of Pearson’s problem – in a model of sound waves travelling though uniform materials.

*Markov property*: if you want to predict the future behaviour of the random walker, you only need know where they are right

*now.*Knowing where they have been in the past adds no helpful insight whatsoever!

*three-dimensional random walks*– like those taken by inebriated birds, or solar photons – there is only about a one-in-three chance of returning to the point of origin. Thus do photons eventually, inevitably, drift free of the sun after a predictable period of time.

## Continuous random walks

“…agreement of the Brownian movement with the requirements of the kinetic hypothesis… justify the most cautious scientist in now speaking of the experimental proof of the atomic nature of matter” – Wilhelm Ostwald, 1909

“Observe what happens when sunbeams are admitted into a building and shed light on its shadowy places. You will see a multitude of tiny particles mingling in a multitude of ways… their dancing is an actual indication of underlying movements of matter that are hidden from our sight.”

*Brownian motion*, and the explanation of its origins would provide the first definitive proof for the existence of atoms.

By modelling Brownian motion as a random walk with tiny, random step sizes, driven by molecular collisions, Einstein provided the mathematics that enabled the very first estimates of the size of individual molecules. Einstein’s equations were experimentally verified by Jean Perrin four years later, finally providing the first conclusive proof for the long-suspected existence of atoms.

Beyond watery pollen, Brownian motion arises in a wide range of circumstances. Perhaps the most widespread is the phenomena of diffusion. Any time you open a perfume bottle, a fresh bag of coffee or any other aromatic container, the pleasant scent that you experience is due to the fragrant molecules being carried all the way from the container to your nose through Brownian-like collisions with the gas molecules in the atmosphere.

In 1923, the American mathematician Norbert Wiener devised a general type of random process, known as a Wiener process, of which Brownian motion is but one specific case. Wiener showed that the paths followed in such processes were both continuous everywhere and differentiable nowhere – in fact, the paths were fractal in nature. And to tie everything together, it became apparent that, when the step sizes were made ever smaller, the majority of discrete random walks became Wiener processes – hence explaining the ubiquity of Brownian motion in nature.

## Walking on π (and other nifty numbers)

“If you are seeking creative ideas, go out walking.” – Raymond Inmon

The mathematics of random walks has recently found a new and very novel application in the analysis of walks taken on numbers, as first described in a 2013 paper by Francisco J. Aragón Artacho, David H. Bailey, Jonathan M. Borwein and Peter B. Borwein (a collection of related publications can be viewed here).

A number like 1/3, which has decimal expansion 0.333333… is not particularly interesting – the walk will keep going in the same direction forever. A walk on the famous circle constant π, whose digits begin 3.141592… is far more fascinating. In the original Aragón, Bailey, Borwein and Borwein paper, the following walk is taken on the first *100 billion* digits of π:

As you can see, this long walk on π bears a striking similarity to a random walk. This is almost certainly not a coincidence – in fact, new pictures such as these may help us resolve a long-standing mathematical question regarding the ‘randomness’ of the digits of π.

Since the ancient Greeks, humanity has been fascinated by the digits of π. In 1761, the Swiss mathematician Johann Lambert proved what had been suspected many centuries prior – that π was an irrational number. That is to say, it cannot be written as a fraction and so its digits go on and on forever without any repeating pattern. Today, another question with a long-suspected but unresolved answer remains regarding whether or not π is a ‘normal number‘ – do the digits 0-9 show up equally often in the decimal expansion of π, or do you eventually reach a point where you’re swamped by 7s (for instance)? That is, do the digits of π behave in a ‘random manner’?

Although we don’t have any proof that the digits of π are behaving randomly, the walks on π are certainly suggestive – and in fact, recent computations have pegged the probability that π is normal (in base 16) at over 99.999… (with more than 3000 other 9s) percent!

Any time random motion is present – be it drifting molecules, fluctuating stock prices or escaping sunlight – the mathematics of random walk theory allows us to extract predictable features from the otherwise unpredictable. And at the current frontiers of mathematical research, it is literally allowing us to see familiar numbers in a whole new light.