Imagine, you end up blindfolded in the middle of a dense, unknown city. At each crossroad, flips of a coin determine your next steps: left, right, forward, or backward. With no vision to guide you and randomness as your only companion, you begin an unpredictable journey.
This, in essence, captures the spirit of random walks, a strong concept from probability theory that’s far more useful than walking through a city blindfolded with a coin in our hand. Physicists use random walks to explain the movement of particles, and have applications in areas starting from biology to social sciences. Understanding random walks allows data scientists to model, simulate and predict stochastic processes from many alternative areas.
Furthermore, in reinforcement learning, agents can perform random walks to explore their environments and gain information in regards to the effects of their actions.
Briefly, random walks are extremely versatile. But that may be a whole other story.
Applications aside, random walks are simply fascinating. Even without the maths behind them, we are able to appreciate the attractive, yet complex and puzzling world they open for us. If you happen to randomly walk around the town long enough and trace your steps, your path reveals a surprising pattern:
The true mystery of random walks emerges when considering different dimensions. Our example of wandering through a city with coin flips is basically a walk in two dimensions: we are able to move forward/backward — the primary 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 would possibly have guessed, for higher-dimensional random walks we’ve got increasingly more directions to pick from. For example, a bird can move left/right, forward/backward, and up/down. If it moves randomly, we’ve got a random walk in three dimensions.
Visualizing random walks of even higher dimensions becomes hard, but we’ll get…
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