Percolation properties of the 2 D Heisenberg model

The most of the problems raised by the authors of the comment [1] about Ref. [2] are based on claims which have not been written in [2], for instance almost all the introduction and the point (1) in [1] are based on such non– existent claims. Instead in Ref. [2] we avoid to make claims not based on well–founded results. For instance in the abstract we write " ... This result indicates how the model can avoid a previously conjectured Kosterlitz–Thouless [KT] phase transition... " and in the conclusive part we notice that " Our results exclude this massless phase for T > 0.5 ". Therefore it seems to us that the opening sentence in the Comment [1] " In a recent letter Allés et al. claim to show that the two dimensional classical Heisenberg model does not have a massless phase. " is strongly inadequate. As for the points that appear in the Comment: • (1) The purpose of the paper [2] is to fill a gap in the research about the critical properties of the Heisenberg model. This gap is the following one: in Ref. [3] a scenario was proposed where the 2D Heisenberg model should undergo a KT phase transition at a finite temperature. This scenario is based mainly on three hypotheses, the third one (which states the non–percolation of the S–type or equatorial clusters) being left in [3] without a plausible justification. To back up that hypothesis a numerical