Long-range models in 1D revisited

In this short note, we revisit a number of classical result{s} on long-range 1D percolation, Ising model and Potts models [FS82, NS86, ACCN88, IN88]. More precisely, we show that for Bernoulli percolation, FK percolation and Potts models, there is symmetry breaking for the $1/r^2$-interaction at large $\beta$, and that the phase transition is necessarily discontinuous. We also show, following the notation of [ACCN88] that $\beta^*(q)=1$ for all $q\geq 1$.