Knowledge of basic probability. A course in measure theory is highly recommended but not strictly required. Mathematical maturity consistent with a first degree in mathematics.
Aim of the course
The aim of this course is to introduce several of the main stochastic processes that are being studied in modern probability theory and to showcase some attractive results.
We plan to cover: random walks and Markov chains, Brownian motion, Poisson and determinantal point processes and several models used in statistical physics such as the Ising model.
At the end of the course the student:
- Is able to give an overview of the basic models used in modern probability theory and their interrelationships.
- Is able to classify Markov chains according to properties such as recurrence/transience, irreducibility and periodicity. Can compute, using transition matrices, various quantities of interest pertaining to Markov chains, such as expected hitting times and exit probabilities.
- Is able to compute quantities of interest for random walks on graphs using elementary potential theory. In particular, can prove Polya's theorem about recurrence and transience of random walks on lattices.
- Can describe the abstract construction of Brownian motion and prove some of the elementary properties of this process, such as continuity of trajectories and the distribution of the maximum of a trajectory.
- Is able to define the Ising model on graphs and explain the underlying formalism involving Gibbs measures. Can compute moments for the one-dimensional model using the transfer matrix method. Can state and explain the phase transition of the model in higher dimension, as well as describe the main steps of the proof of this transition.
- Is able to derive basic theorems on points processes such as Campbell's theorem and Mecke's formula.