Long-range models in 1D revisited

In this short note, we revisit a number of classical results on long-range 1D percolation, Ising model and Potts models (Comm. Math. Phys. 84 (1982) 87-101; Comm. Math. Phys. 104 (1986) 547-571; J. Stat. Phys. 50 (1988) 1-40; Comm. Math. Phys. 118 (1988) 303-336). More precisely, we show that for Bernoulli percolation, FK percolation and Potts models, there is symmetry breaking for the 1/r2-interaction at large 0, and that the phase transition is necessarily discontinuous. We also show, following the notation of (J. Stat. Phys. 50 (1988) 1-40) that 0*(q) = 1 for all q >= 1.