Georgia Institute of Technology College of Sciences

The concentration of measure phenomenon is of great importance in probabilistic combinatorics and theoretical computer science. For example, in the theory of random graphs, we are often interested in showing that certain random variables are concentrated around their expected values. In this talk we consider a variation of this theme, where we are interested in proving that certain random variables remain concentrated around their expected trajectories as an underlying random process (or random algorithm) evolves. In particular, we shall give a gentle introduction to the differential equation method popularized by Wormald, which allows for proving such dynamic concentration results. This method systematically relates the evolution of a given random process with an associated system of differential equations, and the basic idea is that the solution of the differential equations can be used to approximate the dynamics of the random process. If time permits, we shall also sketch a new simple proof of Wormalds method.