The correlation function gives the response function (comes from the fluctuation-dissipation theorem: check it out). When the response function diverges, one can have spontaneous ordering, as happens in the case of magnetization, where we have spontaneous ordering when the response function diverges. Why this happens is obvious from the following equation:

m = XB

where m is the magnetization and B is the magnetic field and X is the response function, known as susceptibility. So when X diverges, one can have finite m even when B is zero.