• definitions of probability
• binomial, Gaussian, and Poisson distributions
• characteristic functions and sums of random variables, Central Limit Theorem,
• random walks

Random number generators 

• randomness in Nature vs codes
• congruential rndm(), overflow problems, rndm-generator period, shuffled rndm()
• transformation method
• generating arbitrary distributions; rejection and hybrid methods
• universal integrals evaluator

• Why using random sampling make sense at all; body volume problem
• ergodicity, balance Eq., detailed balance Eq., Metropolis algorithm
• single spin-flip algorithm for the Ising model; heat bath update
• convergence, errorbars, blocking, bootstrap and jackknife methods
• autocorrelation function and thermalization time
• quantities to calculate, estimators
• correlation functions, Fast Fourier Transform    
• reweighing, histogram, and multiple histogram methods

• continuous spin  XY- and  O(N)-models
• q-state Potts model
• glasses
• first-order transitions
• conserved order-parameter Ising model and the classical lattice gas
• polymers
• kinetic equations

From “simple” to “art”: more efficient and elaborate methods

• What a better data structure can do?
• Swendsen-Wang, Wollf and Niedermayer cluster algorithms
• invaded cluster algorithm
• worm algorithm (WA)
• high-temperature expansions for various models and WA
• WA for polymers
• entropic sampling; Wang-Landau approach
• simulated tempering, parallel tempering

Quantum Monte Carlo

• quantum-to-classical maping; Feynman representation of Quantum mechanics
• using path-integrals in discrete and continuous spaces; quantum configuration space
• sign problem
• diagrammatic expansions, Feynman diagrams

• path-integral MC in continuous space
• diagrammatic MC; how it works for the interacting lattice systems
• s-wave scattering problem example
• Worm algorithm for quantum models
• MC estimators in quantum systems
• Stochastic series expansion
• Variational MC
• Diffusion/projection MC