Lattice models
Summary
Lattice models consist of (typically random) objects living on a periodic graph. We will study some models that are mathematically interesting and representative of physical phenomena seen in the real world.
Content
We will discuss some classical lattice models, such as: random walks, percolation, Ising model, random spanning trees, gaussian free field.
We will prove non-trivial theorems for each of the models. The goal is to allow students to learn general methods and concepts from a number of detailed case studies.
Keywords
probability, graph theory, complex analysis, lattice models, statistical mechanics
Learning Prerequisites
Required courses
Basic probability, basic analysis, linear algebra
While the class will be completely rigorous, the emphasis is more on revealing some interesting phenomena (that somehow exists in nature) rather than on constructing some theories. The goal is to learn things that are generalizable, but I almost always prefer to work out particular cases first.
Recommended courses
None of this is mandatory, but it could help: complex analysis, basic graph theory, simulations
Learning Outcomes
- Reason with probabilistic lattice models
- Manipulate random variables in geometric settings
- Manipulate discrete and continuous objects
Assessment methods
Written exam
Dans le cas de l'€™art. 3 al. 5 du réglement de section, l'€™enseignant décide de la forme de l'€™examen qu'€™il communique aux étudiants concernés
In the programs
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Lattice models
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Lattice models
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Lattice models
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Lattice models
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional