Sommerfeld Theory Colloquium, The hard-disk model has exerted outstanding influence on computational

physics and statistical mechanics. Decades ago, hard disks

were the first system to be studied by Markov-chain Monte Carlo

methods and by molecular dynamics. It was in hard disks, through

numerical simulations, that a two-dimensional melting transition was

first seen to occur even though such systems cannot develop long-range

crystalline order. Analysis of the system was made difficult

by the absence of powerful simulation methods. In recent years, we

have developed powerful Monte Carlo algorithms for hard disks and

related systems. I will in particular show how the event-chain Monte

Carlo algorithm has allowed us to prove that hard disks melt with a

first-order transition from the liquid to the hexatic and a continuous

transition from the hexatic to the solid. I will finally describe how a

new factorized Metropolis filter transforms the event-chain algorithm

into a paradigm for general Monte Carlo calculations. First results

with the generalized algorithm have allowed us to establish the phase

diagram for two-dimensional soft disks and Yukawa particles.