– Pólya’s theorem on recurrence versus transience of simple random walk on Z^d. Green's function. The spectral radius of the d-regular tree.
– Fekete's subadditive lemma, with three applications: return probabilities; the connective constant and the speed of random walks on infinite transitive graphs.
– Chernov's and Azuma-Hoeffding large deviations inequalities. Comparison with the Strong Law of Large – Numbers and the Central Limit Theorem.
– Stochastic domination and couplings.
– Erdős-Rényi random graph phase transitions: subgraph containment, connectivity, giant cluster. First and second moment methods. Critical Galton-Watson trees die out: integer-valued Chung-Fuchs theorem for the recurrence of the exploration random walk.
– Basics of network science: clustering coefficient, isoperimetric ratio (or Cheeger constant), centrality measures: eigenvector centrality, PageRank.
– The basics of Markov chain mixing times: spectral and coupling methods.
– The Barabási-Albert preferential attachment graph, and its degree distribution.
– Renewal processes: SLLN, renewal equations, renewal theorems, renewal paradox, size-biasing.
– Blow-up vs null-recurrence vs positive recurrence in a G/G/1 queuing system.
– Copulas of multivariate continuous distributions.
– Percolation theory: definitions and their equivalence. Examples: p_c(\Z)=1, p_c(\T_d)=1/(d-1), p_c(\Z_2) \in [1/3,2/3] using the Peierls contour method.
– The Ising model on finite graphs: definition, spatial Markov property, basic properties of the partition function, definition of long range order. The Curie-Weiss phase transition.