In the macroscopic world the motion of an object is deterministic.
The fact, this ball is immersed in the atmosphere is relevant in determining its trajectory at short distances.
However, if this ball was trillion time smaller, smaller than a cell; then its trajectory will become stochastic due to the collision with the surrounding air molecules.
This stochastic behaviour is the realm of random walks.
The goal of this tutorial is to outline some elementary, fundamental and beautiful properties of random walks.
I will begin by showing some examples of random walks in nature to highlight their ubiquity and their importances in wide range of phenomena.
Then, I turn to quantitative discussion of basic properties of random walks
I will first show the root mean-square displacement of a random walk grows as a square root of elapse time.
Next, I will discuss the crucial role of spatial dimension on the fundamental question of whether or not a random walk eventually returns to its staring point.
A major portion of this tutorial is devoted to determining the underlying probability distribution of a random walk.
I will also show, how one recovers diffusion equation as continuous limit of the evolution equation for the probability distribution function.
A striking feature of a random walk is that, its probability distribution in long time is independent of almost all microscopic details.
This universality is embodied by the central limit theorem, which I will also present
I will also discuss the enormous features that arise when the very malconditions underlie the central limit theorems are not satisfied.
Finally, I will present some basic first-passage properties of random walks and finish by presenting number of elementary and important applications