Lecturer: Oleg Zaboronski

Term(s): Term 2

Commitment: 30 lectures

Assessment: Oral exam

Timetable: see Additional Resources

Prerequisites: Basic measure theory, Markov chains, discrete stochastic processes. Brownian motions and stochastic analysis might come in handy towards the end of the course. All the prerequisites can be learned from Richard Durret's book [1].

Content: This is a brand new course and it will evolve as I teach it. Your input as the first cohort's will be particularly important. The material presented will be a coherent proper subset of the following:

-The fundamentals of continuous time Markov chains on non-countable configuration spaces (construction, well-posedness theorems, the generator formalism, coupling and duality, analysis of invariant measures and convergence to invariant measures. Working examples: voter models, contact processes, exclusion models.

-Introduction to point processes (definition of simple point processes, correlation functions, Janossi densities, gap probabilities, determinantal and Pfaffian point processes,

-Fredholm determinants and Pfaffians, Szego-type theorems for asymptotic computation of functional determinants. Working examples: Poisson point processes, one-dependent point processes, point processes appearing in random matrix theory, random polynomials

- Reaction-diffusion models in one dimension (annihilating coalescing random walks, branching-coalescing random walks, complete sets of duality functions, extended Pfaffian point processes.

-Large deviations results for the gap size distributions. Continuous limits: point sets for the Brownian Web and the Brownian Net.

References:

[1] Durrett, Rick. Probability: theory and examples. Vol. 49. Cambridge university press, 2019.

[2] Swart, Jan M. "A course in interacting particle systems." arXiv preprint arXiv:1703.10007 (2017).

[3] Liggett, Thomas Milton, and Thomas M. Liggett. Interacting particle systems. Vol. 2. New York: Springer, 1985.

[4] Liggett, Thomas M. Stochastic interacting systems: contact, voter and exclusion processes. Vol. 324. springer science & Business Media, 1999.

[5] Anderson, Greg W., Alice Guionnet, and Ofer Zeitouni. An introduction to random matrices. No. 118. Cambridge university press, 2010.

[6] Daley, Daryl J., and David Vere-Jones. An introduction to the theory of point processes: volume I: elementary theory and methods. Springer New York, 2003.

[7] Schertzer, Emmanuel, Rongfeng Sun, and Jan M. Swart. "The Brownian web, the Brownian net, and their universality." Advances in disordered systems, random processes and some applications (2017): 270-368.

Additional Resources