This course is aimed at Ph.D. students from different fields who expect to use discrete probability in their research. Graduate level measure theoretic probability will be useful, but not a requirement. I expect the course will be accessible to advanced undergraduates who have had sufficient exposure to probability.

We shall illustrate some important techniques in studying discrete random structures through a number of examples. The techniques we shall focus on will include (if time permits)

the probabilistic method;

first and second moment methods, martingale techniques for concentration inequalities;

coupling techniques, monotone coupling and censoring techniques;

correlation inequalities, FKG and BK inequalities;

Fourier analysis on hypercube, Hypercontractivity, noise sensitivity and sharp threshold phenomenon;

Steinâ€™s method;

entropy and information theoretic techniques.

We shall discuss applications of these techniques in various fields such as Markov chains, percolation, interacting particle systems and random graphs.

Suggested books and references:

Noga Alon and Joel Spencer, The Probabilistic Method
,Wiley, 2008.

Geoffrey Grimmett, Probability on Graphs
,Cambridge University Press, 2010.

Ryan O'Donnell, Analysis of Boolean Functions
,Cambridge University Press, 2014.