Central Limit Theorem, extensions and limitations. Large deviations. Limit theorems for extreme values. Tail correlations and Copulas. The law of the iterated logarithm. Sign processes. Record statistics. Survival probabilities. Rare events, optimal fluctuations and hopeful monsters. Examples.
The exponentially large but exponentially rare. Critical points. Urn models and persistence. Multiplicative (GARCH-like) models. Kesten processes. Growth and redistribution processes. Merging processes. Word of mouth models. Growing networks. Simon's process.
Examples and main results: from Gaussian to Gumbel through Lévy. The REM (statics): random condensation transition and replica symmetry breaking (RSB). A toy-model of pinning: baby Functional Renormalisation Group, Burgers equation, shocks and RSB. Intermittent response: a mean-field model for domain wall dynamics. The REM (dynamics): trap models and aging. Ultrametric dynamics.
Gaussian potentials in N-dimensional spaces: short-range vs. long range. Self-affine landscapes. Thermodynamics and dynamics of a particle in a random potential. From non-Arrhenius diffusion to logarithmic diffusion. The N=1 Sinai case. The large N limit; exact solution and the saddle-minima demixing phenomenon. The log-correlated case and the REM transition. Non Gaussian potentials. Multiscale log-correlated potentials and Parisi landscapes. Temperature as a microscope. Multifractal measures and multifractal random walks, with applications.
The model and some of its various incarnations (optimisation, Burgulence, proliferation, etc.). Directed polymers in random media (DPRM) on trees: optimisation, propagating fronts, and some old friends again. DPRM in N-dimensions: review of various techniques (Burgers, Bethe-Kardar, self-consistent resummation, FRG, variational, Derrida-Griffiths, etc) and open problems. Phenomenological description: droplets, intermittent response,disorder/temperature chaos. Variations on the theme (non Gaussian noise and non universality, correlations, etc).
Wigner semi-circle: cavity and free convolutions. Wishart correlation matrices. Student matrices. Lévy matrices. Tracy-Widom distribution, generalisation to heavy tails. Other extreme value matrix models: the SK ground state; generalisations.