Random walks are seemingly simple mathematical problems with far reaching consequences. In this report I will discuss simple random walks and few non-intuitive results which we can derive upon their treatment using combinatorial techniques. We will mostly stick to classical combinatorics and some preliminary analysis techniques to examine certain situations. Towards the end of our treatment of simple random walks we will revisit the gambler's ruin problem.
In the second part of the project I studied multi dimensional Random Walks. Define a lattice as a collection of all the points whose coordinates are integers in \(d\)-dimensional Euclidean space. Let a point \(S_d(n)\) move randomly on this lattice, \(d\) denotes the dimension, and \(n\) denotes the epoch. The rules governing it's motion are that it starts at the origin at time \(0\). At time \(t-1\) if it's at \(S\), then at \(t\) it will be at one of the \(2d\) possible lattice points, with a probability of \(\frac{1}{2d}\). Now, we've two cases. If, \(d\leqslant2\), the walk will return to \(0\) with a probability of \(1\), and it'll happen infinitely often. If, \(d>2\) then the walk will wander off to infinity with probability \(1\). The problems we'll be focusing on in this part are (i) the number of different points through which the walk passes, and (ii) the rate at which a point performing a random walk in \(d\)-space \((d\geqslant3)\) escapes to infinity.
The project report can be viewed or downloaded here.
Prof Anup Biswas (Dept. of Mathematics, IISER Pune)