“…Canonical choices of non-negative clouds are W D C½ K for some compact set K R d (say, a centred ball) containing the origin and for some C 2 .0; 1, or W some non-negative continuous function with compact support, or W.x/ D Cjxj q for some C 2 .0; 1/ and q 2 .0; 1/. Recently there was also some efforts to study the PAM under much less regularity assumptions [Che14], where the potential V is not even a function, but only a measure, and see Example 1.21 for the uncorrelated case. However, one must be careful, as, for W.x/ D Cjxj q with q Ä d, the potential V is infinite almost everywhere (i.e., V Á 1), almost surely [ x i / i , one can also pick the random field .x i / i to be a Gibbsian point field, i.e., a point field that, unlike a Poisson process, has some nontrivial correlation between the particles.…”