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Roughly at the time (1987) when the manuscript for the first three chapters of the present book was completed, several breakthroughs occurred. They had a profound influence on the scope of Monte Carlo simulations in statistical physics, particularly for the study of phase transitions in lattice models. The first of these remarkable developments is the invention of the “cluster algorithms”[4.1–4.54] that reduce (or eliminate completely) the problem of “critical slowing down”(divergence of relaxation times near the critical point, see eg,(2.85),(2.86)[2.59]) from which the single spin-flip algorithms suffer, since the increase of relaxation time means a dramatic increase of statistical errors; see, eg,(2.87). The original version [4.1], proposed for the Ising model, is based on the mapping [4.55] between Potts models [2.106] and percolation [2.60], but meanwhile extensions exist to a wide variety of models, including also isotropic …