Short Courses in Probability Theory

Various Speakers: Planar random growth and scaling limits

14th September, 2020

An LMS online lecture course in random growth.

Conformal growth models are motivated by some real-world growth processes, and are constructed using conformal maps. We will introduce the one-parameter Hastings–Levitov model, which is used to describe Laplacian growth and allows us to vary between off-lattice versions of many well studied models. Then we investigate the "small particle" scaling limit, which often entails finding a martingale and relating its behaviour to its analogue for the proposed continuum limit.

Chak Hei Lo: Foster–Lyapunov methods for Markov chains

10th September, 2020

An LMS online lecture course in Markov chains.

We will start the course by presenting various results using the semimartingale approach for Markov chains. These results include Foster–Lyapunov criteria by which a suitable Lyapunov function can determine whether a process is transient or recurrent. We will then move on to some applications on these methods, including to some random walks on strips and some interacting particles systems, such as voter models.

   1.  Irrational rotations on torus;
   2.  Diophantine approximation: Dirichlet theorem, Roth's theorem, Baker's theory of linear forms of logarithms;
   3.  Furstenberg's ×2,×3 theorem;
   4.  Results and problems on digit expansions of integers;
   5.  Furstenberg's theorem on 2-dimensional torus (if time permits).

Note: For 2., I will mostly state the results without giving proofs as they are out of the scope of this mini-course.