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Random Walks Are Strange and Beautiful

#Random #Walks #Strange #Beautiful

A journey through dimensions and life

Marcel Moosbrugger
Towards Data Science
Photo by Jezael Melgoza on Unsplash

Imagine, you find yourself blindfolded in the center of a dense, unknown city. At each crossroad, flips of a coin decide your next steps: left, right, forward, or backward. With no vision to guide you and randomness as your only companion, you start an unpredictable journey.

This, in essence, captures the spirit of random walks, a powerful concept from probability theory that is much more useful than walking through a city blindfolded with a coin in our hand. Physicists use random walks to describe the movement of particles, and have applications in areas ranging from biology to social sciences. Understanding random walks allows data scientists to model, simulate and predict stochastic processes from many different areas.

Moreover, in reinforcement learning, agents can perform random walks to explore their environments and gain information about the effects of their actions.

In short, random walks are extremely versatile. But that is a whole other story.

Applications aside, random walks are simply fascinating. Even without the math behind them, we can appreciate the beautiful, yet complex and puzzling world they open for us. If you randomly walk around the city long enough and trace your steps, your path reveals a stunning pattern:

A random walk in two dimensions

The real mystery of random walks emerges when considering different dimensions. Our example of wandering through a city with coin flips is essentially a walk in two dimensions: we can move forward/backward — the first dimension — and left/right — the second dimension.

For a one-dimensional random walk, picture an ant walking on a string, taking any step forward or backward with equal probability. Now, as you might have guessed, for higher-dimensional random walks we have more and more directions to choose from. For instance, a bird can move left/right, forward/backward, and up/down. If it moves randomly, we have a random walk in three dimensions.

Visualizing random walks of even higher dimensions becomes hard, but we will get…