### Stochastic Models

(Gábor Pete, 2019)
- Pólya’s theorem on recurrence versus transience of Z
^{d}. Green's function. The spectral radius of the d-regular tree.
- Three notions of amenability: von Neumann, Følner, Kesten. Paradoxical decompositions, Ponzi pyramid scheme. The easy direction of the Kesten-Cheeger theorem: a Følner sequence implies that the Markov operator has norm 1. An example of an amenable Cayley graph with exponential volume growth.
- Fekete's subadditive lemma, with three applications: return probabilities; the connective constant and the speed of random walks on infinite transitive graphs.
- Total variation mixing time of finite Markov chains. Coupling definition of the TV distance, and using it for upper bounds on the mixing time.
- The spectrum of finite reversible Markov chains, with some examples. Upper and lower bounds on the mixing time using the relaxation time.
- Percolation theory: definitions of critical density, basic examples, Peierls contour method, Harris-FKG inequality.
- The Ising model on finite graphs: definition, spatial Markov property, basic properties of the partition function, definition of long range order.